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Posted by Chester Morton / Tuesday, 19 June 2018 / No comments

Statistics or the collection and interpretation of data

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<b><i>This material was taken from a book written by Worlasi Djokoto</i></b> - WALASMATICS

Statistics or the collection and interpretation of data

This material was taken from a book written by Worlasi Djokoto - WALASMATICS

Home MATHEMATICS

Posted by Chester Morton / Sunday, 17 June 2018 / No comments

Standard form or scientific notation

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<b><i>This material was taken from a book written by Worlasi Djokoto</i></b> - WALASMATICS<br />
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Standard form or scientific notation

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The easiest way to learn fractions

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<i><b>This material was taken from a book written by Worlasi Djokoto </b></i>- WALASMATICS

$The easiest way to learn fractions$ The easiest way to learn fractions

This material was taken from a book written by Worlasi Djokoto - WALASMATICS

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Posted by Chester Morton / Monday, 24 April 2017 / No comments

Understanding the basics of sets

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<h1>
<b><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 12.0pt;">SETS</span></b><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> </span></h1>
<h1 style="text-align: justify;">
<span style="font-weight: normal;"><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">A
set is a collection of well-defined objects.<span style="mso-spacerun: yes;"> 
</span>For example a Mathematical Set. It contains Mathematical Instruments
e.g. a Ruler, Template, a Pair of divider, a Pair of Compass, Sets Square etc.
What they have in common is that, they are all used in drawing. Another example
is the cutlery set. This includes Knife, Spoon and Fork, which we use on the
dining table.</span></span></h1>
<h2 align="left" style="text-align: left;">
<span style="font-weight: normal;"><span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">Ç</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> <span style="mso-spacerun: yes;"> </span>is a sign of intersection of sets.</span></span></h2>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">È</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> <span style="mso-spacerun: yes;"> </span>is a sign of unity or union.</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">{}
<span style="mso-spacerun: yes;"> </span>or </span><span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">f</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> denotes empty or null set.</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">Ì</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> <span style="mso-spacerun: yes;"> </span>is a sign of subset.</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">É</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> <span style="mso-spacerun: yes;"> </span>is a sign of super set.</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">h</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> (A) is a
sign that describes the number of elements of set A.</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">Î</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> is a sign
of belonging to a set or a member of a set.</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">Ï</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> is a sign
of not belonging to a set or not a member of a set</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">U
or E. denotes universal set and it is defined as the totality of all sets under
consideration.</span></div>
<h2>
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> </span></h2>
<h2>
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">TYPES
OF SET</span></h2>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Equal
Sets: Equal sets as the name implies are two or more sets having the same
elements in them. For instance the set A = {1, 2, 4} is equal to the set B =
{2, 1, 4}.</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"><span style="mso-tab-count: 1;">            </span></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Disjoint
Sets: Two or more sets are said to be disjoint if they do not have any element
in common. For instance X = {1, 3, 5} and Y = {2, 4, 6} are disjoint sets.</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"><span style="mso-tab-count: 1;">            </span></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Joint
or Intersection Sets: Two or more sets are said to be joint if at least one
element is common to them or at least one element in one is found in the other.
For instance the set Z = {1, 2, 3, 4} and Y = {1, 3, 5, 6} are joint sets
because they have two elements in common i.e. 1 and 3. Thus 1 and 3 become the
point of intersection of the two sets.</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"><span style="mso-tab-count: 1;">            </span></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Subset:
If all elements of a set belong to another set, then we can say that the first
set is a subset of the second. e. g. the set A = {1, 2, 3} is a subset of the
set B = {1, 2, 3, 4, 5, 6}. </span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Note
that an empty set is a subset of every set and a set is a subset of itself i.e.
the set </span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Y
= {2, 4, 6} is a subset of itself.</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"><span style="mso-tab-count: 1;">            </span></span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">To
get the number of subsets of a set, which has n elements, we use this formula 2<sup>n</sup>.
</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">For
example a set of 2 elements will have 2<sup>2</sup> i.e. 2 x 2 = 4 subsets.</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal" style="margin-left: 72.0pt; text-indent: -72.0pt;">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Example1.1<span style="mso-tab-count: 1;">     </span> If U = {0, 1, 2}, list all the subsets of
U. <span style="layout-grid-mode: line;">There are 3 elements in the set hence
the number of subset = 2<sup>n</sup>, where n is the number of elements in the
set i.e. </span></span></div>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">n
= 3. So we have 2<sup>3</sup> = 2 x 2 x 2 = 8 i.e. there are 8 subsets. They
are { }, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2} and {0, 1, 2}.</span></div>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;"> </span></span></div>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Null
or Empty Set:<span style="mso-spacerun: yes;">  </span>Any set that has no
element in it is a null set or an empty set and it is denoted by the sign </span><span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">f</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> or {}, Empty set is
a subset of itself and any other set.</span></div>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"><span style="mso-tab-count: 1;">            </span></span></div>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Complementary
Set: This is normally defined with respect to a universal set. If the set A is
a subset of a universal set U, then all the elements in the universal set which
can not be found in the set A become the complementary set of A which is
denoted A<sup><span style="layout-grid-mode: line;">/</span></sup> e.g. if X =
{2, 4, 6} is a subset of U = {1, 2, 3, 4, 5, 6} then the set X<sup><span style="layout-grid-mode: line;">/</span></sup> = {1, 3, 5}.<span style="mso-tab-count: 1;">  </span></span></div>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Example1.2<span style="mso-tab-count: 1;">     </span>The
universal set U = {2, 3, 5, 7}, P = {2, 5} and Q = {5, 7}, <span style="mso-tab-count: 3;">                         </span></span></div>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Find: <span style="mso-tab-count: 1;">   </span>i.<span style="mso-tab-count: 1;">          </span>(P</span><span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">Ç</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Q)<sup> /</sup> <span style="mso-tab-count: 1;">           </span>ii.<span style="mso-tab-count: 1;">          </span>
P<sup>/ </sup></span><span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">È</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> Q<sup>/</sup></span></div>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"><span style="mso-spacerun: yes;">  </span><span style="mso-tab-count: 2;">                      </span><span style="mso-tab-count: 1;">            </span>iii.<span style="mso-tab-count: 1;">         </span>State
the relationship between i and ii.</span></div>
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<br /></div>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Solution:<span style="mso-tab-count: 1;">          </span>i.<span style="mso-tab-count: 1;">          </span>P </span><span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">Ç</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">
Q means the elements common to both sets </span></div>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"><span style="mso-spacerun: yes;">  </span><span style="mso-tab-count: 3;">                                  </span>P </span><span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">Ç</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> Q = {5}</span></div>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">(P
</span><span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">Ç</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> Q)<sup> /</sup> means the elements in the universal sets but not in<span style="mso-spacerun: yes;">                                                                   
</span>the set P</span><span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">Ç</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Q </span><span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">Þ</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> (P </span><span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">Ç</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Q)<sup> /</sup> = {2, 3, 7}.</span></div>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"><span style="mso-spacerun: yes;">   </span><span style="mso-tab-count: 2;">                     </span>ii.<span style="mso-tab-count: 1;">          </span>P<sup>/</sup> = {3, 7} and Q<sup>/</sup> = {2, 3}</span></div>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"><span style="mso-spacerun: yes;">      </span><span style="mso-tab-count: 2;">                  </span><span style="mso-spacerun: yes;">  </span><span style="mso-tab-count: 1;">          </span>P<sup>/
</sup>U Q<sup>/</sup> = {2, 3, 7}</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"><span style="mso-spacerun: yes;">      </span><span style="mso-tab-count: 2;">                  </span>iii.<span style="mso-tab-count: 1;">         </span>The relationship between i and ii is they are equal sets.</span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"></span></div>
<h2 align="left" style="text-align: left;">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> </span></h2>
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<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Example
1.3<span style="mso-tab-count: 1;">    </span>If μ = {3, 6, 9, 12, 15, 18, 21,
24, 27}, N = {3, 6, 9, 12, 15} and </span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"><span style="mso-tab-count: 2;">                        </span>M = {3, 9, 27}. Find (N </span><span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">È</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> M)<sup>/</sup> where
N, M </span><span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">Î</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> μ.</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;">Solution<span style="mso-tab-count: 1;">           </span>N </span><span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">È</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> M = {3, 6, 9, 12, 15, 27}</span></div>
<div class="MsoNormal">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"><span style="mso-tab-count: 2;">                        </span>(N </span><span lang="EN-US" style="font-family: "symbol"; font-size: 11.0pt;"><span style="mso-char-type: symbol; mso-symbol-font-family: Symbol;">È</span></span><span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"> M)<sup>/</sup> =
{18, 21, 24}</span></div>
<div class="MsoNormal">
<br /></div>
<h2 align="left" style="text-align: left;">
<span lang="EN-US" style="font-family: "arial" , "sans-serif"; font-size: 11.0pt;"><br /></span></h2>

Understanding the basics of sets

SETS A set is a collection of well-defined objects. For example a Mathematical Set. It contains Mathematical Instruments e.g. a...

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Posted by Chester Morton / Friday, 27 May 2016 / No comments

Algebraic Expressions

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<span style="font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"></span><b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB">ALGEBRAIC EXPRESSION </span></u></b></span></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">In
algebra, letters are used to represent numbers. Letters that are used to
represent numbers are called variables. Letters which have fixed values are
called constants.<span style="mso-spacerun: yes;">   </span>For example; in the
formulae of area of a circle (πr<sup>2</sup>), r is a variable (it varies) and
π</span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"> </span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">is
the constant (it is always the same).</span></div>
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<br /></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"></span>An algebraic expression is an expression that
contains only letters, numbers and at least one of the operations (+,-, ×, ÷). For
example; 4x – 3y + 6.</span></div>
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<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">NOTE:<span style="mso-spacerun: yes;">  </span>multiplication and division signs are not
usually seen in algebraic expressions, but you have to know that <b style="mso-bidi-font-weight: normal;">ab</b> means a is being multiplied by b,
and <b style="mso-bidi-font-weight: normal;">a/b</b> means that b is dividing a. </span></div>
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<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">In
an example like 2x + 8y – 4, 2 and 8 are the coefficients of x and y
respectively. 2x, 8y and 4 are each called a term separated by an operation. 4
is called a constant term. Please note that the coefficient of any letter
standing alone is 1. <b style="mso-bidi-font-weight: normal;">For example:</b> a
+4a, since the coefficient of a is 1, therefore the final answer is 5a. </span></div>
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<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="text-decoration: none;"><br /></span></span></u></b></div>
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<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">FORMING
AN ALGEBRAIC EXPRESSION</span></u></b></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Statements
in words are often written as algebraic expressions in mathematics. Any letter
may be used for an unknown number, but different letters must be used for each
unknown number.</span></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Examples: Let x represent an unknown
number,</span></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">18 more than the number<span style="mso-spacerun: yes;">    </span><span style="mso-spacerun: yes;"> </span>= x +
18</span></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">23 less than the number<span style="mso-spacerun: yes;">   </span><span style="mso-spacerun: yes;">  </span><span style="mso-spacerun: yes;"> </span><span style="mso-spacerun: yes;">  </span>= x –
23 </span></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">5 times the number<span style="mso-spacerun: yes;">                </span>= 5x</span></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Quarter of the number<span style="mso-spacerun: yes;">          </span><span style="mso-spacerun: yes;"> </span>= x/4</span></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">If 4 times a number is subtracted
from 3 and the result is multiplied by 7<span style="mso-spacerun: yes;"> 
</span><span style="mso-spacerun: yes;"> </span>=7(3 – 4x)</span></div>
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<br /></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Trial
Questions</span></b></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">One less than x</span></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">A certain number plus eight is equal
to three times the number.</span></div>
<div class="MsoNormal">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Half a certain number is three more
than the number.</span></div>
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<br /></div>
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<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="text-decoration: none;"></span></span></u></b></div>
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<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">OPERATIONS
ON ALGEBRAIC EXPRESSION</span></u></b></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">The rules of addition, subtraction, multiplication
and division can be used to simply algebraic expressions.</span></div>
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<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Addition
and Subtraction</span></u></b></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Only like terms can be added or
subtracted to give a single term, this is often called collecting of like
terms.</span></div>
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<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Illustrative
examples</span></u></b></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Simplify the following expressions;</span></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">5x + 4 – 9y + 3x + 2y
– 7 </span></div>
<div class="MsoListParagraphCxSpLast" style="mso-list: l14 level1 lfo1; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">2p – 3q + 6p + 7q</span></div>
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<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Solution</span></b></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">5x + 4 – 9y + 3x + 2y
– 7<span style="mso-spacerun: yes;">  </span></span></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Grouping like terms, we have</span></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">         </span>5x + 3x – 9y + 2y + 4 – 7 </span></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Simplifying like terms, we have </span></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">        </span>8x – 7y – 3 </span></div>
<div class="MsoListParagraph" style="mso-list: l12 level1 lfo2; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">2p – 3q + 6p + 7q</span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Grouping like terms, we have</span></div>
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<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">       </span>2p + 6p
– 3q + 7q</span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Simplifying like terms, we have </span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">        </span>8p +
4q</span></div>
<div class="MsoNormal" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Trial Questions</span></b></div>
<div class="MsoListParagraphCxSpFirst" style="mso-list: l6 level1 lfo3; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">8x + 5y + 4 – 4x + 3y – 2 </span></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l6 level1 lfo3; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">6m + 7n + 5m – 9n</span></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l6 level1 lfo3; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">5q<sup>2 </sup>+ 6q + 4q<sup>2 </sup>+ 7q</span></div>
<div class="MsoListParagraphCxSpLast" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Multiplication
and Division</span></u></b></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">It is very important to first of all group all numbers
and same letters together and then apply basic rules of indices.</span></div>
<div class="MsoListParagraphCxSpFirst" style="margin-left: 66.75pt; mso-add-space: auto; mso-list: l2 level1 lfo4; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">a<sup>m </sup>× a<sup>n </sup>=a<sup>m + n</sup></span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 66.75pt; mso-add-space: auto; mso-list: l2 level1 lfo4; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">a<sup>m</sup></span></u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">    </span>=a<sup>m – n </sup></span></div>
<div class="MsoListParagraphCxSpLast" style="margin-left: 66.75pt; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">a<sup>n</sup></span></div>
<div class="MsoNormal" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Examples under
multiplication:</span></b></div>
<div class="MsoListParagraphCxSpFirst" style="margin-left: 54.75pt; mso-add-space: auto; mso-list: l16 level1 lfo5; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">6<sup>2 </sup>× 6<sup>4</sup></span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 54.75pt; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">6<sup>2+4</sup></span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 54.75pt; mso-add-space: auto; text-align: justify;">
<sup><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">    </span></span></sup><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">6<sup>6</sup></span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 54.75pt; mso-add-space: auto; mso-list: l16 level1 lfo5; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">5a<sup>2 </sup>× 4a</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 54.75pt; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Grouping
the numbers and letters together, we have</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 54.75pt; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">5 × 4a<sup>2
</sup>× a</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 54.75pt; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">20a<sup>2+1</sup></span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 54.75pt; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">20a<sup>3</sup></span></div>
<div class="MsoListParagraphCxSpLast" style="margin-left: 54.75pt; mso-add-space: auto; text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"></span></b></div>
<div class="MsoNormal" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Trial Questions</span></b></div>
<div class="MsoListParagraphCxSpFirst" style="margin-left: 54.75pt; mso-add-space: auto; mso-list: l8 level1 lfo6; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">3x<sup>2 </sup><span style="mso-spacerun: yes;"> </span>× <span style="mso-spacerun: yes;"> </span>4x<sup>3</sup></span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 54.75pt; mso-add-space: auto; mso-list: l8 level1 lfo6; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">2a × 5a<sup>2</sup></span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 54.75pt; mso-add-space: auto; mso-list: l8 level1 lfo6; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">3xy<sup>2 </sup>× 4x<sup>3</sup>y<sup>4</sup></span></div>
<div class="MsoListParagraphCxSpLast" style="margin-left: 54.75pt; mso-add-space: auto; text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Examples under
division:</span></b></div>
<div class="MsoListParagraphCxSpFirst" style="margin-left: 90.75pt; mso-add-space: auto; mso-list: l13 level1 lfo7; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">6<sup>2 </sup>÷ 6<sup>4</sup></span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 90.75pt; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">6<sup>2-4</sup></span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 90.75pt; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">6<sup>-2</sup></span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 90.75pt; mso-add-space: auto; mso-list: l13 level1 lfo7; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">32a<sup>4</sup>b<sup>2 </sup>÷ 4a<sup>2</sup>b</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 90.75pt; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">It
advisable to solve using this procedure </span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 90.75pt; mso-add-space: auto; text-align: justify;">
<u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">32a<sup>4</sup>b<sup>2</sup></span></u></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 90.75pt; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"> </span>4a<sup>2</sup>b</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 90.75pt; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">8a<sup>4-2</sup>b<sup>2-1</sup></span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 90.75pt; mso-add-space: auto; tab-stops: 193.5pt; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">8a<sup>2</sup>b<span style="mso-tab-count: 1;">                           </span></span></div>
<div class="MsoListParagraphCxSpLast" style="margin-left: 90.75pt; mso-add-space: auto; tab-stops: 193.5pt; text-align: justify;">
<br /></div>
<div class="MsoNormal" style="tab-stops: 193.5pt; text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Trial Questions</span></b></div>
<div class="MsoListParagraphCxSpFirst" style="margin-left: 90.75pt; mso-add-space: auto; mso-list: l17 level1 lfo8; tab-stops: 193.5pt; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">16a<sup>2</sup>b<sup>4<span style="mso-spacerun: yes;">  </span></sup>÷ 8ab<sup>3</sup></span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 90.75pt; mso-add-space: auto; mso-list: l17 level1 lfo8; tab-stops: 193.5pt; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">27a<sup>5</sup>b<sup>2<span style="mso-spacerun: yes;">  </span></sup>÷<span style="mso-spacerun: yes;">  </span>9a<sup>2</sup>b</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 90.75pt; mso-add-space: auto; mso-list: l17 level1 lfo8; tab-stops: 193.5pt; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">4a<sup>2 </sup><span style="mso-spacerun: yes;"> </span>÷ 4a<sup>2</sup></span></div>
<div class="MsoListParagraphCxSpLast" style="margin-left: 90.75pt; mso-add-space: auto; tab-stops: 193.5pt;">
<br /></div>
<div class="MsoNormal" style="tab-stops: 193.5pt;">
<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">EXPANDING
EXPRESSIONS </span></u></b></div>
<div class="MsoNormal" style="tab-stops: 193.5pt; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Brackets are used to group two or more terms
together. The process of simplifying such an expression is called <b style="mso-bidi-font-weight: normal;">expanding the expression. </b>The <b style="mso-bidi-font-weight: normal;">law of distribution</b> is applied, where
the outside term multiples each and every term in the bracket.<b style="mso-bidi-font-weight: normal;"> Be very careful with the operational signs
when expanding expressions.</b></span></div>
<div class="MsoNormal" style="tab-stops: 193.5pt;">
<br /></div>
<div class="MsoNormal" style="tab-stops: 193.5pt;">
<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Illustrative
Examples:</span></u></b></div>
<div class="MsoListParagraphCxSpFirst" style="margin-left: 1.5in; mso-add-space: auto; mso-list: l11 level1 lfo9; tab-stops: 193.5pt; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">2a(6a + 7b)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 1.5in; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">2a
multiplying each term, we have</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 1.5in; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">(2a ×
6a) + (2a × 7b)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 1.5in; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">12a<sup>2
</sup>+ 14ab</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 1.5in; mso-add-space: auto; mso-list: l11 level1 lfo9; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">m(2p – 4q + 2r)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 1.5in; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">m
multiplying each term, we have</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 1.5in; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">(m ×
2p) – (m × 4q) + (m × 2r)</span></div>
<div class="MsoListParagraphCxSpLast" style="margin-left: 1.5in; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">2mp – 4mq
+2mr</span></div>
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"></span></b><br />
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"></span>Trial Questions</span></b>

<br />
<div class="MsoListParagraphCxSpFirst" style="margin-left: 1.5in; mso-add-space: auto; mso-list: l10 level1 lfo10; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">– 3 (2x – 4)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 1.5in; mso-add-space: auto; mso-list: l10 level1 lfo10; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">3a(4a + 2b + ab)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 1.5in; mso-add-space: auto; mso-list: l10 level1 lfo10; text-align: justify; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">9(4m – 3n)+3(2n + m)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 1.5in; mso-add-space: auto; text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"></span></div>
<div class="MsoListParagraphCxSpLast" style="margin-left: 1.5in; mso-add-space: auto; text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">MULTIPLYING
TWO BINOMIALS</span></u></b></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">A binomial is an expression, consisting of two
terms.<b style="mso-bidi-font-weight: normal;"> For example:</b> (2q+p) and (m+n)
are all binomials. When multiplying two binomials, it is necessary to use the
law of distribution. That is each term in the first bracket will multiply each
term in the other bracket.</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Illustrative Example:</span></u></b><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"></span></u></div>
<div class="MsoListParagraphCxSpFirst" style="margin-left: 51.0pt; mso-add-space: auto; mso-list: l1 level1 lfo11; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">(2a – b)(a + 2b)</span></div>
<div class="MsoListParagraphCxSpLast" style="margin-left: 51.0pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">    </span>2a (a + 2b)
– b (a + 2b)</span></div>
<div class="MsoNormal">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">                       </span>2a<sup>2 </sup>+ 4ab – ab
– 2b<sup>2</sup></span></div>
<div class="MsoNormal">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">                      </span>= 2a2<sup> </sup>+ 3ab –
2b<sup>2</sup></span></div>
<div class="MsoListParagraphCxSpFirst" style="margin-left: 51.0pt; mso-add-space: auto; mso-list: l1 level1 lfo11; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">(3x + 2y)(3x – 2y)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 51.0pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">    </span>3x (3x – 2y)+2y(3x – 2y)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 51.0pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">        </span>9x<sup>2</sup> – 6xy + 6xy – 4y<sup>2</sup><span style="mso-spacerun: yes;">  </span></span></div>
<div class="MsoListParagraphCxSpLast" style="margin-left: 51.0pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">           </span>9<sup>x2</sup>
– 4y<sup>2</sup></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Trial Questions</span></b></div>
<div class="MsoListParagraphCxSpFirst" style="margin-left: 51.0pt; mso-add-space: auto; mso-list: l4 level1 lfo12; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">(4x + y)(2x + 3y)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 51.0pt; mso-add-space: auto; mso-list: l4 level1 lfo12; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">(a - b)(2a - 2b)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 51.0pt; mso-add-space: auto; mso-list: l4 level1 lfo12; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">(4m - 3n)(2m + n)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 51.0pt; mso-add-space: auto; mso-list: l4 level1 lfo12; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">4.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">(6p + 2q)(3p - q)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 51.0pt; mso-add-space: auto; mso-list: l4 level1 lfo12; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">5.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">(4p + 4q)(2s + 3t)</span></div>
<div align="center" class="MsoListParagraphCxSpLast" style="margin-left: 51.0pt; mso-add-space: auto; text-align: center;">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">FACTORIZING AN ALGEBRAIC EXPRESSION</span></u></b></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">When each term in an expression has a common
letter or number, we divide each term by the common number or letter and
enclose the quotient in a bracket; the factor is placed outside the bracket.
The process of putting an expression in bracket is called<b style="mso-bidi-font-weight: normal;"> FACTORIZATION. For example: </b>6x+6y have 6 to be its highest common
factor. Hence, we have the final answer to be 6(x + y).</span></div>
<div class="MsoNormal">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">There are
three method of solving algebraic expressions;</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Method 1:<span style="mso-spacerun: yes;"> 
</span>Common Factor Method</span></u></b><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">In this method, you will have to find a factor
which is common to all the terms in the expression and bring it outside the bracket,
hence leaving the uncommon factors in the bracket.</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><b style="mso-bidi-font-weight: normal;"><u>Illustrative
Examples</u></b></span></div>
<div class="MsoListParagraphCxSpFirst" style="margin-left: 38.25pt; mso-add-space: auto; mso-list: l5 level1 lfo13; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">100x – 25x<sup>2</sup></span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 38.25pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">In this expression,
the highest common factor is 25x </span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 38.25pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"> </span>25x (4 – x)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 38.25pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"> </span>We can expand the answer to be sure its
correct.</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 38.25pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">25x (4 – x)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 38.25pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">(25x × 4) – (25x × x)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 38.25pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">100x – 25x<sup>2</sup></span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 38.25pt; mso-add-space: auto;">
<br /></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 38.25pt; mso-add-space: auto; mso-list: l5 level1 lfo13; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">5ax+25ay</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 38.25pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">The highest common
factor is 5a, hence</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 38.25pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">  </span>5a(x+5y)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 38.25pt; mso-add-space: auto;">
<br /></div>
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Trial Questions</span></b>

<br />
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 38.25pt; mso-add-space: auto; mso-list: l9 level1 lfo14; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">2×5.89<span style="mso-spacerun: yes;">  </span>+ 2×3.11</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 38.25pt; mso-add-space: auto; mso-list: l9 level1 lfo14; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">4x + 2x<sup>2</sup></span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 38.25pt; mso-add-space: auto; mso-list: l9 level1 lfo14; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">3a<sup>2</sup>x + b<sup>2</sup></span></div>
<div class="MsoListParagraphCxSpLast" style="margin-left: 38.25pt; mso-add-space: auto;">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Method 2: Grouping Method </span></u></b></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">In this type, there are usually four terms. You
have to group them into two binomials, and then find the common factor of each
binomial and bring the common factor outside the bracket of each binomial.</span></div>
<div class="MsoNormal" style="text-align: justify;">
<br /></div>
<div class="MsoNormal" style="text-align: justify;">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">NOTE:</span></b><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"> You can rearrange the terms to ensure that each
binomial has a common factor within.</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Illustrative Examples:</span></u></b><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"></span></u></div>
<div class="MsoListParagraphCxSpFirst" style="margin-left: 74.25pt; mso-add-space: auto; mso-list: l15 level1 lfo15; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">ab – by – ay + y<sup>2</sup></span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 74.25pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">(ab – by) – (ay + y<sup>2</sup>)
</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 74.25pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"> </span>b(a – y) – y(a – y<sup> </sup>)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 74.25pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">(b – y) (a – y)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 74.25pt; mso-add-space: auto; mso-list: l15 level1 lfo15; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">3x<sup>2<span style="mso-spacerun: yes;">  </span></sup>+ 2xy – 12xz –
8yz </span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 74.25pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">(3x<sup>2 </sup>+
2xy) – (12xz – 8yz)</span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 74.25pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">x (3x + 2y) – 4z(3x +
2y)</span></div>
<div class="MsoListParagraphCxSpLast" style="margin-left: 74.25pt; mso-add-space: auto;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">   </span>(x – 4z) (3x + 2y)</span></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Trial Questions</span></b></div>
<div class="MsoListParagraphCxSpFirst" style="margin-left: 74.25pt; mso-add-space: auto; mso-list: l0 level1 lfo16; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">x<sup>2 </sup>– xm + xy – my </span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 74.25pt; mso-add-space: auto; mso-list: l0 level1 lfo16; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">9r<sup>2</sup> + 3rs – 15rt – 5st </span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 74.25pt; mso-add-space: auto; mso-list: l0 level1 lfo16; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">4p<sup>2</sup> – 6tp + 18pq + 27tq </span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 74.25pt; mso-add-space: auto;">
<br /></div>
<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Method 3: Quadratic Trinomials</span></u></b>

<br />
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Trinomial is an expression with three terms. Any
expression in the form ax<sup>2</sup>+bx+c, where a, b, and c are constants are
not equal to zero is a quadratic trinomial.</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><b style="mso-bidi-font-weight: normal;">For example:</b></span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"> </span>2x<sup>2 </sup>+
3x +1 is a quadratic trinomial where 2 and 3 are the coefficient of x<sup>2</sup>
and x respectively, and 1 is a constant.</span></div>
<div class="MsoNormal" style="text-align: justify;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">There are steps involved in solving a quadratic
equation or expression;</span></div>
<div class="MsoListParagraphCxSpFirst" style="mso-list: l7 level1 lfo17; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     </span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Multiply the coefficient of x<sup>2</sup> by the constant.</span></div>
<div class="MsoListParagraphCxSpMiddle" style="mso-list: l7 level1 lfo17; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Find the possible pair of factors of the results in step 1, which when
added gives the coefficient of x.</span></div>
<div class="MsoListParagraphCxSpLast" style="mso-list: l7 level1 lfo17; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Replace the coefficient of x by the factors obtained in step 2 and use
the method of grouping to complete the factorization.</span></div>
<br />
<b style="mso-bidi-font-weight: normal;"><u><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Illustrative
example:</span></u></b>

<br />
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Example 1</span></b><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">: 4x<sup>2</sup> – 8x + 3</span></div>
<div class="MsoListParagraphCxSpFirst">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Step 1:</span></b><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"> The coefficient of x<sup>2</sup> is 4 which is
multiplied by the constant, which is 3. Therefore we have, 4 × 3=12</span></div>
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<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Step 2:</span></b><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"> Possible pair of factors of 12 are (1 and 12), (2
and 6), (3 and 4), (-2 and -6), and a few other. Now, (-2 and -6) are the only
factors that adds up to -8 which is the coefficient of x.</span></div>
<div class="MsoListParagraphCxSpLast">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Step 3:</span></b><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"> Replace -8x by -2x -6x in the expression and use
grouping method to factorize.</span></div>
<div class="MsoNormal">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">          </span>4x<sup>2</sup> – 8x + 3 = 4x<sup>2</sup>
– 2x – 6x + 3</span></div>
<div class="MsoNormal">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">                                  </span>2x (2x – 1) –
3(2x – 1)</span></div>
<div class="MsoNormal">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">                                  </span>(2x – 3) (2x
– 1)</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Example 2: </span></b><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;"> </span>3x<sup>2</sup>
– 2x – 5 </span></div>
<div class="MsoNormal">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">             </span><b style="mso-bidi-font-weight: normal;">Step 1: </b>The coefficient of x<sup>2</sup> is 3 which is multiplied
by the constant, which is -5 in this case. Therefore we have, 3 × (-5) = -15</span></div>
<div class="MsoNormal">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">             </span><b style="mso-bidi-font-weight: normal;">Step 2:</b> The<b style="mso-bidi-font-weight: normal;"> </b>possible
pair of factors of – 15 is (- 1 and 15), (1 and – 15), (– 3 and 5) and (5 and –
3). But of these, only the pair of (3 and – 5) adds up to – 2 which is the
coefficient of x. </span></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">           
</span>Step 3:</span></b><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">
Replace – 2x by 3x and – 5 xs in the expression and use the grouping method to
factorize.</span></div>
<div class="MsoNormal">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">3x<sup>2</sup>
– 2x – 5 =3x<sup>2</sup> + 3x – 5x – 5 </span></div>
<div class="MsoNormal">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">                       </span>3x(x + 1) – 5(x + 1)</span></div>
<div class="MsoNormal">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">                       </span>(3x – 5)(x + 1)</span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<b style="mso-bidi-font-weight: normal;"><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">Trial Questions </span></b></div>
<div class="MsoListParagraphCxSpFirst" style="margin-left: 39.75pt; mso-add-space: auto; mso-list: l3 level1 lfo18; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">1.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">X<sup>2</sup> + 6x – 27 </span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 39.75pt; mso-add-space: auto; mso-list: l3 level1 lfo18; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">2.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">5x<sup>2</sup> – 13x – 6 </span></div>
<div class="MsoListParagraphCxSpMiddle" style="margin-left: 39.75pt; mso-add-space: auto; mso-list: l3 level1 lfo18; text-indent: -.25in;">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-list: Ignore;">3.<span style="font: 7.0pt "Times New Roman";">     
</span></span></span><span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;">X<sup>2</sup> – 8x + 12</span></div>
<div class="MsoListParagraphCxSpLast" style="margin-left: 39.75pt; mso-add-space: auto;">
<br /></div>
<div class="MsoNormal">
<span lang="EN-GB" style="font-family: "calibri light" , "sans-serif"; font-size: 12.0pt; line-height: 107%;"><span style="mso-spacerun: yes;">  </span></span></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoNormal">
<br /></div>
<div class="MsoListParagraph">
<br /></div>
<div class="MsoNormal" style="margin-left: .5in;">
<br /></div>
<div class="MsoNormal">
<br /></div>

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