C h a p t e r 3 A l g e b r a i c o p e r a t i o n s 11331 �3u + tv + ut � 3v 32 x2 � 1 33 12x2 � 7x + 1

34 (x � 1)2 � 4 35 (x + 2)2 � 16 36 (2x + 3)2 � 25

37 3(x + 5)2 � 27 38 25 � (x � 2)2 39 4(3 � x)2 � 16y2

40 (x + 2y)2 � (2x + y)2 41 (x + 3)2 � (x + 1)2

42 (2x � 3y)2 � (x � y)2 43 (x + 3)2 + 5(x + 3) + 4

44 (x � 3)2 + 3(x � 3) � 10 45 2(x + 1)2 + 5(x + 1) + 2

1 Expand and simplify 4(x � 1)(2x � 3). 2 Expand �7(x + 3)2.

3 Expand (2x � 7)(2x + 7). 4 Factorise 24x3 � 18x.

5 Factorise 98x2 � 72y2. 6 Factorise 4x2 � 8x + xy � 2y.

7 Factorise x2 � 9x � 22. 8 Factorise 6x2 + 19x + 15.

9 Factorise 4x2 � 26x � 14. 10 Factorise x2 + 6x + 9.

Simplifying algebraic fractionsIn this section we look at using factorisation techniques to simplify more complexalgebraic fractions. After factorising a numerator or denominator, terms may becancelled, resulting in a simplified fraction, or even no fraction at all.

Always look for common factors in the numerators and denominators beforecancelling.

WorkSH

EET 3.2

1

Simplify each of the following fractions by factorising the numerator and denominatorand cancelling as approriate.

a b

Continued over page

THINK WRITE

a Write the fraction. a

Factorise both numerator and denominator. =

Cancel any common factors (4). =

4x 8+12

---------------- x2 3x+

x2 9–------------------

14x 8+

12---------------

24 x 2+( )

12--------------------

3x 2+

3------------

14WORKEDExample

114 M a t h s Q u e s t f o r W e s t e r n A u s t r a l i a B o o k 3

If there is a quadratic trinomial in either the numerator, the denominator or both, thenfactorise before cancelling.

THINK WRITE

b Write the fraction. b

Factorise both numerator and denominator. =

Cancel any common factors (x + 3). =

1x2 3x+

x2 9–-----------------

2x x 3+( )

x 3+( ) x 3–( )----------------------------------

3x

x 3–-----------

Simplify each of the following algebraic fractions by first factorising the numerator and denominator.

a b c

THINK WRITE

a Write the numerator, making sure that the expression is in the correct order.

a x2 + 3x � 4

Find the factor pair of c (�4) which adds to b (3).

Factor sum of �4: �1 + 4 = 3

Write the expression and its factorised form. x2 + 3x � 4 = (x � 1)(x + 4)

Write the original fraction.

Replace the numerator with the factorised form.

=

Cancel common factors and simplify if appropriate.

= x + 4

b Write the numerator, making sure that the expression is in the correct order.

b x2 � 7x � 8

Find the factor pair of c (�8) that adds to b (�7).

Factor sum of �8: 1 � 8 = �7

Write the expression in its factorised form, using the appropriate factor pair.

x2 � 7x � 8 = (x + 1)(x � 8)

Write the denominator, making sure that the expression is in the correct order.

x2 + 3x + 2

Find the factor pair of c (2) that adds to b (3).

Factor sum of 2: 1 + 2 = 3

x2 3x 4–+x 1–

--------------------------- x2 7x– 8–x2 3x 2+ +--------------------------- x2 6x– 5+

2x2 16x– 30+-------------------------------------

1

2

3

4x2 3x 4–+

x 1–--------------------------

5x 1–( ) x 4+( )

x 1–----------------------------------

6

1

2

3

4

5

15WORKEDExample

C h a p t e r 3 A l g e b r a i c o p e r a t i o n s 115

When multiplying or dividing algebraic fractions, it is often necessary to factorisebefore cancelling common factors. When dividing, remember to change the divisionsign to a multiplication sign and take the reciprocal of the second fraction (invert thesecond fraction).

THINK WRITE

Factorise using the appropriate factor pair. x2 + 3x + 2 = (x + 1)(x + 2)

Write the original fraction.

Replace the numerator and denominator with the factorised form.

=

Cancel any common factors and simplify if appropriate.

=

c Write the numerator, making sure that the expression is in the correct order.

c x2 � 6x + 5

Find the factor pair of c (5) that adds to b (�6).

Factor sum of 5: �1 + �5 = �6

Factorise using the appropriate factor pair. x2 � 6x + 5 = (x � 1)(x � 5)Write the denominator, making sure that the expression is in the correct order and take out the common factor of 2.

2x2 � 16x + 30 = 2(x2 � 8x + 15)

Find the factor pair of c (15) that adds to b (�8).

Factor sum of 15: �3 + �5 = �8

Factorise using the appropriate factor pair. 2x2 � 16x + 30 = 2(x � 3)(x � 5)

Write the original fraction.

Replace the numerator and denominator with the factorised form.

=

Cancel any common factors and simplify if appropriate.

=

6

7x2 7x– 8–

x2 3x 2+ +---------------------------

8 x 1+( ) x 8–( )x 1+( ) x 2+( )----------------------------------

9 x 8–x 2+------------

1

2

34

5

6

7x2 6x– 5+

2x2 16x– 30+------------------------------------

8x 1–( ) x 5–( )

2 x 3–( ) x 5–( )-------------------------------------

9x 1–

2 x 3–( )--------------------

Simplify each of the following by factorising the numerator and denominator and cancelling as appropriate.

a � b ÷

Continued over page

8x2 16x+8

------------------------- 6x 12–

x2 4–------------------ 4x

3x2 2x– 5–------------------------------ 4x 12–

3x2 3x+----------------------

16WORKEDExample

116 M a t h s Q u e s t f o r W e s t e r n A u s t r a l i a B o o k 3

THINK WRITE

a Write the expression. a �

Factorise each numerator and denominator as appropriate.

= �

Cancel any common factors; one in any numerator and one in any denominator. The factors that can be cancelled are 8, (x + 2) and (x � 2).

= �

Multiply the numerators and multiply the denominators.

=

Simplify if necessary. = 6x

b Write the expression. b ÷

Change the ÷ sign to � and invert the second fraction.

= �

Factorise each numerator and denominator as appropriate.

= �

Cancel any common factors; one in any numerator and one in any denominator. The factors that can be cancelled are 4 and (x + 1).

= �

Multiply the numerators and multiply the denominators.

=

This cannot be simplified any further.

18x2 16x+

8------------------------ 6x 12–

x2 4–------------------

28x x 2+( )

8----------------------- 6 x 2–( )

x 2+( ) x 2–( )----------------------------------

38x x 2+( )

8----------------------- 6 x 2–( )

x 2+( ) x 2–( )----------------------------------

46x1

------

5

14x

3x2 2x– 5–----------------------------- 4x 12–

3x2 3x+---------------------

24x

3x2 2x– 5–----------------------------- 3x2 3x+

4x 12–---------------------

34x

3x 5–( ) x 1+( )------------------------------------- 3x x 1+( )4 x 3–( )-----------------------

44x

3x 5–( ) x 1+( )------------------------------------- 3x x 1+( )4 x 3–( )-----------------------

53x2

3x 5–( ) x 3–( )-------------------------------------

6

When simplifying algebraic fractions:1. factorise the numerator and the denominator2. cancel factors where appropriate3. if 2 fractions are multiplied, factorise where possible then cancel any factors,

one from the numerator and one from the denominator4. if 2 fractions are divided, remember to multiply the reciprocal of the second

fraction before factorising and cancelling.

remember

C h a p t e r 3 A l g e b r a i c o p e r a t i o n s 117

Simplifying algebraic fractions

1 Simplify each of the following by factorising the numerator and denominator andcancelling as appropriate.

a b c

d e f

g h i

j k l

m n o

p

2 Simplify each of the following algebraic fractions by first factorising the numerator andthe denominator.

a b c

d e f

g h i

j k l

m n o

p

3 Simplify each of the following by factorising the numerator and denominator andcancelling as appropriate.

a � b � c ÷

3FSkillSHEET

3.12

Simplificationof algebraic

fractions

WORKEDExample

14

5x 10+15

------------------ 6x4x 2+--------------- x2 x+

x--------------

Mathcad

Simplifyingalgebraicfractions

3x2 6–6

----------------- x2 1–x 1–-------------- 2x2 2–

x 1+-----------------

x2 x+

x2 x–-------------- x

3x2 x–----------------- x2 5x–

x2 25–-----------------

4x2 8+

4x2----------------- 3x2 3–

3x2 3+----------------- x2 9–

x2 3x+-----------------

x2 1–

7x2 7x+--------------------- x2 4x–

x2 16–----------------- 2x 2–

4x2 4–-----------------

3x2 6x–

6x2 3x–--------------------

WORKEDExample

15x2 5x 6+ +

x 3+--------------------------- x2 7x 12+ +x 4+------------------------------ x2 9x– 20+

x 5–-----------------------------

b2 8b– 7+b 1–

--------------------------- a2 18a– 81+a 9–

--------------------------------- a2 22a– 121+a 11–

------------------------------------

x2 49–x 7–

----------------- m2 64–m 8–

------------------ a2 7a– 12+a2 16–

------------------------------

p2 4 p– 5–p2 25–

--------------------------- x2 6x 9+ +x2 2x 3–+--------------------------- m2 2m– 1+

m2 5m 6–+-----------------------------

y2 4y– 12–y2 36–

----------------------------- x2 4x– 4+x2 4–

-------------------------- x2 3x 40–+x2 6x 16–+-----------------------------

x2 3x 18–+x2 6x– 9+

-----------------------------

WORKEDExample

16

SkillSHEET

3.13

Multiplicationof fractions

SkillSHEET

3.14

Divisionof fractions

6x2 12x+3

------------------------ 2x 4–

x2 4–--------------- 5x2 15x–

20----------------------- 4x 12+

x2 9–------------------ 2x 4+

x2 x– 6–----------------------- 2x

x2 3x–-----------------

118 M a t h s Q u e s t f o r W e s t e r n A u s t r a l i a B o o k 3

d ÷ e � f ÷

g ÷ h �

i � j �

k � l �

Equal or not equal?Algebraic terms have a different value depending on the value assigned to the pronumeral. Some algebraic terms, however, will be equal regardless of the value of the pronumeral.

Are the expressions and equal for all values of x?

1 From our knowledge of numbers, we know that any number divided by itself will be equal to 1. The same will apply to algebraic terms. If two algebraic terms or expressions are equal, the result of division of one by the other will be 1.

Perform the division by first fully factorising the expression.

2 Are the two expressions equal?

3 Check your answer by substituting the value x = 2 into each expression.

4 What happens when you attempt to substitute x = 1 into each expression?

What’s the problem?Each expression below has been factorised, but incorrectly! Identify what the problem is with each and then factorise the expression correctly. (If it cannot be factorised, write ‘Cannot be factorised’ as your answer.)

1 x2 + 7x – 8 = (x – 8)(x + 1)

2 x2 + 16 = (x + 4)(x + 4)

3 3x2 – 6x + 3 = 3x(x – 6)

4 x2 + 6x + 5 = (x + 2)(x + 3)

x2 x– 6–

x2 5x 6+ +--------------------------- 2x 6–

6x 9+--------------- 3x 6–

x 6+--------------- x 6+4x 8–--------------- x2 9–

x2 1–-------------- 3x 9–

3x 3+---------------

x2 5x– 6+

x2 x– 6–-------------------------- x2 x 6–+

x2 5x 6+ +--------------------------- m2 8m 15+ +

m2 7m 10+ +--------------------------------- m2 6m 8+ +

m2 10m 21+ +------------------------------------

x2 6x 8+ +x2 5x 6+ +--------------------------- x2 8x 15+ +

x2 7x 12+ +------------------------------ x2 x 6–+

x2 5x 14–+----------------------------- x2 9x 14+ +

x2 x– 12–------------------------------

y2 y 20–+y2 7y 10+ +------------------------------ y2 y– 6–

y2 3y– 4–-------------------------- x2 7x– 6+

x2 x 2–+-------------------------- x2 2x– 8–

x2 x– 12–--------------------------

x 5+x2 x+-------------- x2 4x 5–+

x3 x–--------------------------

x 5+x2 x+-------------- x2 4x 5–+

x3 x–--------------------------÷