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Sim
plif
ying
Alg
ebra SIMPLIFYING
ALGEBRA
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Simplifying Algebra
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J 15
Algebra is mathematics with more than just numbers. Numbers have a fixed value, but algebra introduces variables – whose values can change. These are represented by letters. Although they work similarly to numbers it is important to be aware of how to add, subtract, multiply and divide expressions containing variables.
SIMPLIFYING ALGEBRA
What do I know now that I didn’t know before?
Answer these questions, before working through the chapter.
I used to think:
Answer these questions, after working through the chapter.
But now I think:
What is the difference between like and unlike terms?
Write an example explaining the distribution property
What is a binomial product?
What is the difference between like and unlike terms?
Write an example explaining the distribution property
What is a binomial product?
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Basics
Like Terms
It’s easy to see that 2 3 6# = and 3 2 6# = , so it doesn't matter in which order 2 and 3 are multiplied. This is always true! In any situation a b b a# #= . This is even true if more than two terms are multiplied: a b c a c b b a c# # # # # #= = .
This means xy = yx and they are like terms. abc = acb = bac are all equal and are also like terms.
The times sign #^ h is dropped in multiplcation of different terms. So x x4 4# = , a b ab# = and x y z xyz2 2# # =
The coefficients are always multiplied separately. Here are some more examples.
Terms with the same variables and indices are called like terms. If indices or variables differ they are called unlike terms.
Examples of like and unlike terms
Like Terms Unlike Terms
and
and
p p
xy xy
2 4
3 7
2 2
2 2
Different vairables
Different indices
andm n5 4
and4 4a b ab2 2
Only like terms can be added or subtracted.
Simplify the following as much as possible
Write the following products using algebra
2 3 7 2 9a ab a ab a ab2 2 2+ + - = +
Like terms
Like terms
This can't be simplified anymore because these are unlike terms.
Multiplying Terms
a b c3m n
mn3
#
=
p q r
p q r
pqr
2 4
2 4
8
# #
# # # #=
=
^ h
xy x y
x x y y
x y
6 3
6 3
18
2 3
2 3
3 4
#
# # # #=
=
^ ^ ^h h h
Coefficients multiplied separately Coefficients multiplied separately
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Basics
Rules for Multiplication
When terms have the same sign their product is positive, when they have different signs their product is negative.
• #+ + = +
• #+ - = -
• #- + = -
• #- - = +
Positive # Positive = Positive
Positive # Negative = Negative
Negative # Positive = Negative
Negative # Negative = Positive
Here are some examples of multiplying with signs
Simplify the following
a
c
a b
b
d
a b
a b
ab
7 2
7 2
14
#
# # #
-
= -
=-
^ ^h h
m n p
m n p
mnp
2 2 5
2 2 5
20
# #
# # # # #
- -
= - -
=
^ ^h h
x y
x y
xy
3 5
3 5
15
#
# # #
- -
= - -
=
^ ^h h
t u v
t u v
tuv
5 4
5 4 1
20
# #
# # # # #
- - -
= - - -
=-
^ ^h h
The middle steps of c and d can be skipped if you remember this rule:
• If there are an even amount of negatives then the product is positive.
• If there are an odd amount of negatives then the product is negative.
Order of Operations
Remember that brackets and multiplication (and division) are always done before addition and subtraction.
Simplify the following as much as possible
xy x y3 2 4#+ -
xy xy3 8= -
xy5=-
Simplify like terms -(-42) = 42
Multiplication FIRST Multiplication FIRST Multiplication FIRST
a b b c5 2 6 7# #- - -
( )ab ac10 42=- - -
10 42ab ac=- +
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Basics
2. Complete the following
1. Say if these are like or unlike terms
3. Use addition and subtraction to simplify the following as much as possible
a and 2a anda a2
pq and qp andcd d c2 2
a
a
a
b
b
b
c
c
c
c
d
d
d
d
3 4 4# #= yz =
ghi = abcd =
4 5 3 1x x x+ - + 11 20 8 7a b b a- + +
2 3 7 5 6jk j k jk k- + + + + 6 2 3 6ab ba ab ba+ - +
4 5 7 2def edf de ef ed- + + + - 4 2 3 8 4x y xy x xy yx2 2 2 2 2+ - + +
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Basics
4. Use multiplication and the order of operations to simplify the following as much as possible
ba
dc
fe
hg
ji
lk
a4 3# x y2 6#
5 5t u#- 10p q#-
7 4 2b d c# #- 3 6g h i# #- -
8 2 3ux v w# #- - - 2 3 2p q r s# # #- -
x x y4 2 2#+ 3 2 4x y xy#- +
a b b a5 2 3 10# #+ - x w w z wz zw7 4 2 2# #- - - + +
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Knowing More
When algebraic terms are divided, algebraic fractions are formed. These can be simplified by cancelling like terms.Write the division as a fraction. Always simplify the coefficients and cancel the variables where necessary.
Algebraic Fractions (Dividing Terms)
Simplify the following as much as possible:
a bm m
mm
mm
3 6
63
63
21
'
=
#=
=
Write as fraction Write as fraction
y is only in the numerator, so it is not cancelled. x is in the numerator and denominator, so x is cancelled
m is in both the numerator and denominator, so it can be cancelled
x xy12 4'
xyx
xyx
y
412
412
3
#
=
=
=
Rules for Division
Algebraic division with signs have the same rules as multiplication: If the terms have the same sign their quotient is positive, when they have different signs their quotient is negative.
• '+ + = + or ++ =+
• '+ - = - or -+ =-
• '- + = - or +- =-
• '- - = + or -- =+
Positive ' Positive = Positive
Positive ' Negative = Negative
Negative ' Positive = Negative
Negative ' Negative = Positive
Remember, always write the division as a fraction first. Then simplify the coefficients and cancel the variables.
a b
Simplify the following as much as possible:
Write as fraction Write as fraction
• The coefficients have been simplified • x is in both the numerator and
denominator, so it can be cancelled• The answer is negative '+ -^ h
• The coefficients have been simplified • a and c are in both the numerator
and denominator, so they can be cancelled
• b cannot be cancelled because it only appears in the numerator
• The answer is positive '- -^ h
x x15 10'- abc ac20 2'- -
xx
1015
23
=-
=-
acabc
b
220
10
=-
-
=
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Knowing More
a b c
Order of Operations
Remember that brackets and division (and multiplication) are always done before addition and subtraction.
Simplify
Simplify
a bx xy y
yxy
x
6 4 2
24
4
'-
=
x
x x
6
6 2
= -
= -
division has been done firstmultiplication and division have been done first
Revising Index Laws
Here is a reminder of index laws.
1. Multiplication with indices: a a am n m n# = +
3. Raising an index to an index: a am n mn=^ h
5. Quotients raised to indices: ba
b
am
m
m
=` j
7. Negative indices: oraa b
aab1n
n
n n= =- -
` `j j
2. Division with indices: a aa
a am nn
mm n
' = = -
4. Products raised to indices: ab a bm m m=^ h
6. The zero index: for1 0a a0 !=
8. Fractional indices: a a anm mn n m
= = ^ h
pqr r p q
rpqr
pq
36 12 7 4
12
36
31
' #+
=
pq
pq pq
28
3 28
= +
= +
Index laws are used to simplify algebraic fractions. More than one law might be necessary to simplify a fraction.
pq p q q
p qq
q
p q
p q
1 1
2 3 3 8 0 4
3 64
4
3 6
3 2
' #
' #=
=
=
-^ ^h h b b
b b
b
b
34 4 7
43
47
48
2
#
#=
=
=
^ hx y x y
x y
x y
xy
y
x
1
2 3 5 7
2 5 7 3
7 4
7
4
4
7
# '
=
=
=
=
+ -
-
c m
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Knowing More
Combining Index Laws and Algebraic Fractions
The next logical step is to mix indices and algebraic fractions. Algebraic fractions with indices work with the same two steps.
• Step 1: Simplify the numerator and denominator separately.
• Step 2: Simplify the fraction and cancel variables where necessary.
Here are some examples of algebraic fraction containing indices.
a
c d
b
Simplify as much as possible:
y y yy y y
yy
y
y
4 2 24 2 2
816
2
2
3
3 1
2
# #
+ +
=
=
=
-
Simplify numerator and denominator separately
Simplify numerator and denominator separately
Simplify numerator and denominator separately
Simplify fraction using index laws for division
Simplify fraction using index laws for division
x is in the denominator because its index is negative
Simplify numerator and denominator separately
Simplify fraction using index laws for division
Simplify fraction using index laws for division
Fractional indices
x y z
x y z
x y z
x y z
x y z
x y z
xy z
18
6
18
36
2
2
2
5 2
2 3 2
5 2
4 6 2
4 5 6 2 2 1
1 4
4
=
=
=
=
- - -
-
^ h
x y
x y
x y
x y
x y
x y
x y
x y
16
12
16
12
4
12
3
3
6 8
6 8
6 8 21
6 8
3 4
6 8
6 3 8 4
3 4
=
=
=
=
- -
^ h
p q
p q p q
p q
p q p q
p q
p q
p q
p q
2 4
8 16
128
128
128
3 3
2 3 2 2 2
3 3
6 3 4 4
3 3
10 7
10 3 7 3
7 4
#
#
-
=-
=-
=-
=-
- -
^ ^h h
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Questions Knowing More
1. Find the quotient of the following:
a
a
e
c
g
d
d
b
f
d
b
e
c
f
a a8 2' a a2' pqr pr4 2'-
100 20amn n'- - wx xy18 9'- st stu5 20'-
2. Answer these questions about order of operations:
Simplify a b4 2#
Simplify a b4 2'
Simplify b a2 4#
Simplify 2b a4'
Is this statement true or false: a b b a4 2 2 4# #=
Is this statement true or false: a b b a4 2 2 4' '=
Is the order of the terms in multiplication important?
Is the order of the terms in division important?
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Knowing MoreQuestions
3. Use the order of operations to simplify these as much as possible:
a
d
b
e
c
f
a ab b10 40 10'- xyz xz y26 2 6' +
xz xyz xz24 4# ' xy x yz z20 5 30 6' '+ d f def e15 3 200 10# '-
14 2 6mn n m' -
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Questions Knowing More
a
c
e
b
d
f
4. Simplify these algebraic fractions as much as possible
y y yy y y2 33 4 5
# #
+ +
x x
x
2 2
42 4
3 2
#^
^
h
h
x y
xy x y
2
5 22 2 3
2 2 2 3#
^
^ ^
h
h h
pq
pq p q32
2 3 3#
-
-^ ^h h
f g f g
f g fg
3 2
9 22 2 4
2 3 3
#
#
^
^
h
h
q r
qr q r
3 2 2
2 3 2#-
^e
ho
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Knowing MoreQuestions
5. Answer these questions
a
b
Use the index law aa
1nn
=- to prove ba
abn n
=-
` `j j (Hint: )ba
ban n1
=- -
` ``j j j
Use this result to simplify this fraction x y
xy
3
32 3
2 2 2-
^
^e
h
ho
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Using Our Knowledge
p q p
pq p p
3 2 6 9
6 18 272
- + -
=- - +
^ h
Expanding Brackets
A term outside a bracket is multipled with all terms inside a bracket, so that
a b c a b a c ab ac# #+ = + = +^ h
This property of multiplication with brackets is called the 'distribution property'. Here are some examples:
Expand the following
Expand the following
Expand the following
a
a
a
b
b
b
x
x
3 2
6 3
+
= +
^ h
( )y
y
4 5
4 20
- -
=- +
y4 # 4 5#- - p q3 2#- 3p p6#- 3p 9#- -
x x y
x x xy
5 2 3
5 2 32
+ +
= + +
^ h
5x # x x2# x y3#3 2#- 3 x#
There is no difference if the number on the outside is negative. Always multiply the term outside with all the terms inside.
Sometimes like terms can be simplified after expanding brackets.
4 3x x x x2 5 22- + -^ ^h h
x x23 10 2= -
x x x x8 4 15 62 2= - + -
Like terms
Like terms Like terms
Like terms
2q p pq q p pq1 4 3 2 5- + - + -^ ^h h
q pq pq pq pq2 8 6 2 102 2=- - + + -
q pq pq2 6 4 2=- - -
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Using Our Knowledge
Multiplying Brackets
Brackets can also be multiplied together. Both terms in the first bracket are multiplied with both terms in the second bracket.
( )( )a b c d+ +
( )( )x x3 2+ + ( )( )y y2 3 3 4+ -
1
1
3
3
4
4
2
2
ac ad bc bd= + + +
The product of two brackets can also be thought of in this way
( )( ) ( ) ( )a b c d a c d b c d
ac ad bc bd
+ + = + + +
= + + +
Find the following products:
a
c
b
d
1 1
1 1
3 3
3 3
4 4
4 4
2 2
2 2
( ) ( ) ( ) ( )x x x x2 3 3 2# # #= + + + + ( ) ( ) ( ) ( )y y y y2 3 2 4 3 3 3 4# # # #= + - + + -
x x x2 3 62= + + + y y y6 8 9 122
= - + -
x x5 62= + + y y6 122
= + -
Like terms
Like terms
p q p q
p p p q q p q q
p pq pq q
p pq q
2 3 4
2 2 4 3 3 4
2 8 3 12
2 5 12
2 2
2 2
# # # #
+ -
= + - + + -
= - + -
= - -
^ ^
^ ^ ^ ^
h h
h h h h
a b c d
a c a d b c b d
ac ad bc bd
3 5 3 2
3 3 3 2 5 3 5 2
9 6 15 10
# # # #
- +
= + + - + -
= + - -
^ ^
^ ^ ^ ^
h h
h h h h
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Questions Using Our Knowledge
1. Expand these brackets:
a
c
e
g
i
b
d
f
h
j
( )y5 2 3+ ( )t6 4 3-
( )m4 3 5- - -
( )x y21 8 2-
( )x3 9 4- -
( 4 )n m41 16 2- -
2x x y4 2-^ h x xy y4 32-^ h
5p q pq p2 3+ +^ h 3xy x y2 2 1- +^ h
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Using Our KnowledgeQuestions
2. Expand these brackets:
a x x y xy2 3 4 4+ - +^ h b 2abc ab ac a b bc c3 4 2 5 32 2- - - + - +^ h
4 2x x x x3 7 52- + -^ ^h h
3. Expand then simplify using like terms:
a
c
b
d
d c cd cd d cd1 2 3 5 3 22 2- + + - -^ ^h h
5 4p q pq q p q p3 2 2 2 32 2- - + -^ ^h h 3 (8 2 4 )ab a b b a b a b ab a a b2 62 2 3 2 2 2
- - + - + - +^ h
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Questions Using Our Knowledge
4. Expand and simplify:
a
c
e
g
b
d
f
h
x x2 2 1+ -^ ^h h st t t s2 3 4 2+ -^ ^h h
k k4 1 3 2- -^ ^h h
x y x y+ -^ ^h h
ab a b ab2 3 62- +^ ^h h
ab cd ab cd- +^ ^h h
x y xy xy x3 4 22 2- +^ ^h h ab c a bc abc a b c3 4 5 23 2 2 2 2 2
- -^ ^h h
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Using Our Knowledge
Step 1: Find a common denominator
Step 2: Write both fractions over this denominator
Step 1: Find a common denominator
Step 2: Write both fractions over this denominator
Adding and Subtracting Algebraic Fractions
Like all fractions, algebriac fractions can only be added or subtracted if they have the same denominator. If the fractions already have a common denominator, then simplify the numerators only.
Find the following products:
a
c
a b
d
b
These fractions already have a common denominator
12 is the lowest common denominator of 3, 4 and 6
30 is the lowest common denominator of 15, 30 and 10
ab is the lowest common denominator of a and ab
This is the product of the original denominators
This is the product of the original denominators
Use the distributive property to expand the bracket
Multiply the brackets to find this denominator
Sometimes its difficult to find a common denominator. In these situations, the common denominator can be found by multiplying the denominators together.
Adding and subtracting fractions with complicated denominators
x 32
75
++
7 x 3+^ h
x x
x
x
x
7 314
7 3
5 3
7 3
14 5 3
=+
++
+
=+
+ +
^ ^^
^^
h hh
hh
Step 3: Simplify Step 3: Simplify
xx
xx
7 2114 5 15
7 215 29
=+
+ +
=++
m n m n2 2+
--
( )( )m n m n+ -
m n m n
m n
m n m n
m n
m n m n
m n m n
2 2
2 2
=+ -
--
+ -
+
=+ -
- - +
^ ^^
^ ^^
^ ^^ ^
h hh
h hh
h hh h
m mn mn n
m n m n
m n
n
2 2 2 2
4
2 2
2 2
=+ - -
- - -
=--
y y y
y y y
y
3 43
6
124 9 2
1211
+ -
=+ -
=
p q p q
p q p q
q q
15
8
30
2
10
2
30
16 2 6 3
30
25
6
5
- ++
=- + +
= =
ab a
abb
5 5
5 5
+
= +
h h
h h
h
199
194
199 4
195
-
= -
=
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Questions Using Our Knowledge
5. Simplify these fractions:
a
c
e
g
b
d
f
h
x x47
45+
x x32
125-
mn km3 2+
x65
1 22+-
a a2 13 4-
-y y3 13
52
++
y y64
11
+-
- p p p
325 2
2+ -
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Multiplying and dividing algebraic fractions is easy because a common denominator is not necessary. To multiply algebraic fractions, simply multiply the numerators and the denominators separately.
Multiplying and Dividing Algebraic Fractions
Find these products:
a
c
a b
b
d
xy
xy
xy
34
43
43
#
#
#=
=
pqp
qp p
qp
3
2
3
2
3
2 2
#
#
#=
=
t
t
t
t
t
t
t
t
510
5
10
5
10
2
2
4
2
2
4
2
4
4 2
2
#
#
#=
=
=
=
-
a b
a b
a b
a b
a b
a b
a b
b
a
2 2
24
2 2
24
4
24
6
6
7 2
4 4
4 4
7 2
4 4
7 2
7 4 2 4
2
3
# #
#
#=
=
=
=
- -
To divide algebraic fractions, simply find the reciprocal of the second fraction and multiply.
Find these quotients:
x y
xy
yx
yx
yx
82
16
82 16
82 16
832
4
'
#
##
=
=
=
=
cab
cb
cab
b
c
c b
ab c
b c
abc
ba
43
2412
53
12
24
5 12
3 24
60
72
56
2
2
2
2
'
#
#
#
-
= -
= -
= -
=-
Multiply numerators and denominators separately
Multiply numerators and denominators separately
'flip' the second fraction to find the reciprocal.
'flip' the second fraction to find the reciprocal.
Multiply numerators and denominators separately
Multiply numerators and denominators separately
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Questions Using Our Knowledge
6. Write these in simplest form:
a
c
e
g
b
d
f
h
p p2 10#
mnn510
#
mn
m n5252 3
#p q
p
q7
3 62
4#
ps t
t
p s9
8
16
273 2
4
3 2
#ba
cb
dc
# #
e f
d
d
e f d ef
5
3
3
443 2 4
4 5 2 3
# #a
wx ywxya b
2
52510
2
2 3 4
#-
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7. Write these in simplest form:
a
c
e
b
d
f
p q2
3
6'
ba a
2
2
#
a b a b3
49
23 5 2 2
#f
d e
d f
e
8
3
16
92
2
2
3
'
x y
s t
x y
s t5 203 4
3 2
6 2
5
'p q r
a b c
p q r
a bc
7
6
21
24 3 2
2 3 2
2 5
3 4
'
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Thinking More
p q
p p q q
p pq q
4 2
4 2 4 2 2
16 16 4
2
2 2
2 2
-
= + - + -
= - +
^
^ ^ ^ ^
h
h h h h
Shortcuts for Multiplying Brackets
A product of two brackets it called a ‘binomial product’. Some binomial products can be found more easily than others. Look at this example:
a b a b
a ab ab b
a b
2 2
2 2
+ -
= - + -
= -
^ ^h h
The middle terms cancel each other away
Can you see the shortcut? Their product is the square of the first term minus the square of the second term. Here are some examples using this shortcut:
Find these products:
x x
x
x
2 2
2
4
2 2
2
+ -
= -
= -
^ ^
^ ^
h h
h h
a b y y
y
y
3 5 3 5
3 5
9 25
2 2
2
- +
= -
= -
^ ^
^ ^
h h
h h
What about the perfect square of a bracket? Look at this example:
2
a b a b a b
a ab ab b
a ab b
2
2 2
2 2
+ = + +
= + + +
= + +
^ ^ ^h h h
1st termsquared
1st termsquared
2nd termsquared
2nd termsquared
2nd termsquared
2 # product of terms
1st termsquared
2nd termsquared
2 # product of terms
1st termsquared
2nd termsquared
2 # product of terms
Can you see the shortcut? Here are some examples using this shortcut:
Find these products:
x
x x
x x
4
2 4 4
8 16
2
2 2
2
+
= + +
= + +
^
^ ^ ^ ^
h
h h h h
a b
1st termsquared
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Thinking More
These shortcuts may seem silly for these easier examples, but they are especially useful for more complicated examples.
Find these products:
a
c d
bx
xx
x
xx
xx
4 4
4
16
2 2
22
+ -
= -
= -
` `
` ^
j j
j h
x y pq x y pq
x y pq
x y p q
3 4 3 4
3 4
9 16
2 3 2 3
2 2 3 2
4 2 2 6
+ -
= -
= -
^ ^
^ ^
h h
h h
2
25 2
25
yy
y yy y
yyy
y
yy
523
5 523
23
215
4
9
154
9
2
22
22
22
-
= + - + -
= + - +
= - +
c
^ ^ c c
c
m
h h m m
m
2
4 2 9
4 2 9
m n n
m n m n n n
m n m n n
m n m n n
2 3
2 2 3 3
6
1
3 2 2
3 2 2 3 2 2
6 4 3 3 2
6 4 3 3 2
-
= + - + -
= + - +
= - +
^
^ ^ ^ ^
^
h
h h h h
h
We can even use these rules in calculations using actual numbers.
a b11 9
11 9 11 9
20 2
40
2 2-
= + -
=
=
^ ^
^ ^
h h
h h
36
30 2 6
900 360 36
30 6
30 6
1296
2
2
2 2
= +
= + +
= + +
=
^
^ ^
h
h h
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Questions Thinking More
1. Prove that x y x y x y2 2+ - = -^ ^h h
2. Simplify these perfect squares
a
c
e
b
d
f
p2 1 2+^ h x y5 7 2-^ h
m n3 2 2- -^ h p pq3 2-^ h
xx1 2
+` j yy
2 322
-c m
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Thinking MoreQuestions
3. Simplify these binomial products:
a
c
e
g
b
d
f
h
x x3 3+ -^ ^h h m m5 5- +^ ^h h
b c b c2 3 2 3+ -^ ^h h x y x y7 3 7 3+ -^ ^h h
xy x y xy x y2 2 2 2+ -^ ^h h a b ab a b ab10 4 10 42 3 2 2 3 2
- +^ ^h h
mm
mm
21
212 2
+ -` `j jx
yx
y52 3
52 3- +` `j j
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Questions Thinking More
4. Find 9 and 4 in each of the following
a
c
a
e
c
b
f
d
b
d
9x x x3 2 2+ = + +9^ h 4 9x x xy y2 122 2 2
+ = + +9^ h
5 16 40 25q p pq q2 2 2- = - +9^ h 4 16m m mn2 2 2 2
+ = + +9 4^ h
5. Find these values using expanded form:
16 62 2- 64 362 2
-
292 372
77 272 2- .12 62 (Hint: 12.6 = 12 + 0.6)
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Thinking MoreQuestions
6. A stage in a stadium is made up of a rectangle and a square in the following diagram. The dimensions are in terms of x.
Find expression for the area of the square in the form of a binomial product
Find expression for the area of the rectangle in the form of a binomial product
Show that the total area of the stage is ( )( ) ( )x x x2 2 4 2 3 2- + + -
Simplify the expression for the area of the stage
Find the value of the area of the stage if 5x =
Rectangle
Square
2x + 4
2x - 3
a
b
c
d
e
x -
2
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Answers
Basics: Knowing More:
Knowing More:
Using Our Knowledge:
a
a
a
b
b
b
c
c
c
e
d
d
d
d
3 4 4 3# #=
x6 1+ a b4 12-
j k jk3 13 3+ + ab11
edf ed f3 3 73+ +
x y xyz x8 10 32 2+ -
Like terms
Like terms Like terms
Unlike terms
yz z y#=
ghi g h i# #=
abcd a b c d# # #=
1.
2.
3.
4.
1.
2.
2.
3.
4.
5.
1.
2.
a
c
g
e
i
k
b
d
h
f
j
l
a12 xy12
tu25- pq10-
bcd56- ghi18
uvwx48- pqrs12
x xy4 4+ xy2-
ab20- zw xw11 7-
a
a
e
c
g
d
h
b
f
d
b
e
c
f
4a1 q2-
am5 yw2-
u41-
8ab
8ab
True, they are both 8ab.
False, they are not equal.
Yes, the answers are different.
No, the answers are the same.
ba2
ab
2
a
d
b
e
c
f
a6 y19 m
xyz6 y9 df25
a
c
b
d
y
22 x2
12
x y25 2 p q9 4 3
e ff
g43
3r81
2
a bba
abn n
=-
` `j jy
x92
8
a
c
e
g
i
b
d
f
h
j
y10 15+ t24 18-
m12 20+
x y4 -
x27 12+
n m4 6- +
x xy8 42- x y yx4 33 2
-
pq p q p5 10 152 2+ +
x y xy xy6 6 32 2- +
a x x xy x y2 3 4 42 2+ - +
b 6 2 8 4 10 6abc a b c a bc a b ab c abc2 2 2 2 3 2 2 2 3- + + - + -
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Answers
x x2 62-a
c
b
d
cd cd d c d cd3 3 3 23 2 2 2 3+ + - -
pq q q7 12 182 2+ -
ab a b a b4 13 113 2 3 2- -
a
c
b
d
x2 22-
st s t t st8 4 12 62 2 2- + -
k k k8 14 3 22 2- + - +
ab a b ab a b6 6 83 2 2 2 2+ - -
e
f
x y2 2-
a b c d2 2 2 2-
g x y x y x y x y3 6 4 83 2 3 2 3 2 2+ - -
h a b c a b c a b c a b c15 6 20 82 4 2 3 5 3 3 3 3 4 3 4- - +
Using Our Knowledge: Using Our Knowledge:
Thinking More:
3.
4.
5.
6.
7.
2.
3.a
a
a
c
c
c
e
e
e
g
b
b
b
d
d
d
f
f
f
h
x3 x4
kmnk n3 2-
xx
6 1217 10
--
a aa2 1
11 4--
^ h y yy
5 3 121 2
++
^ h
y yy1 6
3 10
- +-
^ ^h h p
p
2
16 42
-
a
c
e
g
b
d
f
h
p20
2
m2
mn
52 p
q
7
182
2
t
p s
2
32
2 5
da
de f5
2 6
a bxy2 2-
a
c
e
b
d
f
qp9
ba2
3
a b21
8 5 7
fe
d
3
22
4
s y
tx
4 2 2
3
ac p r
b q92 2
2 2
p p4 4 13+ +
x xy y25 70 492 2- +
m mn n9 12 42 2+ +
p p q p q9 62 2 2 2+ +
xx1 2
2
2+ +
y yy
4 12 94
2- +
g
h
x 92- m25 2
-
b c4 92 2- x y49 92 2
-
x y x y2 2 4 4-
a b a b100 164 6 2 4-
mm
4
12
4-
x xy
xy25
456
56 9
2 - + -
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Answers
Thinking More:
4.
5.
6.
a
c
b
d
6=9 y3=9
p4=9 n
n
4
16
2
4
=
=
9
4
a
e
c
b
f
d
220 2800
841 1369
5200 .158 76
a
e
b
d
x2 3 2-^ h
x x2 2 4- +^ ^h h
x x6 6 12- -
119
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Notes