algebraic fractions - MadAsMaths · Created by T. Madas Created by T. Madas Question 2 Simplify the...

Created by T. Madas

ALGEBRAIC

FRACTIONS

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SIMPLIFYING

RATIONAL

EXPRESSIONS

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Question 1

Simplify the following algebraic fractions.

x −,

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Question 2

Simplify the following algebraic fractions.

− −

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Question 3

Simplify the following algebraic fractions:

− −

5tt−+

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Question 4

− −

2 3z+, 5

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Question 5

8 39 5

− −

12 36 15

5 14 3

( )2 5

2 52 1

+, 2 1

2 5 1w

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Question 6

d) 3 2

2 13 6

+ − +

, 2 2 43 2

−, 5

6w− , 2 1t +

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Question 7

d) ( )( )2

2 1 3 6

6 19 9 10

− + +

( )2 32 1x

2 2 43 2

−, 5

6w− , 3

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ADDING

ALGEBRAIC

FRACTIONS

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Question 1

Simplify the following expressions into a single fraction.

a) 1 2

b) 4 1

3x x−

3 6tt+ +

d) ( ) ( )1 1

2 42 3

w w+ + −

Created by T. Madas

Question 2

Simplify the following expressions into a single fraction.

a) 1 3

2 1x x+

b) 4 3

1 2y y−

c) 2 1

2 1 3t t−

d) ( ) ( )

2 1 4 1w w+

( )( )

+ −,

( )( )

2 1 3t t+ +,

( )( )

Created by T. Madas

Question 3

Simplify the following into a single fraction.

a) 2 3

1 2x x+

b) 5 4

1 2y y−

c) 4 2

2 1 3t t−

( )( )

+ −,

( )( )

+ −,

( )( )

2 1 3t t− +,

( )( )

Created by T. Madas

Question 4

Simplify the following algebraic expressions giving your final answer as a single

fraction in its simplest form.

a) ( )( )

1 1 2 1x x x−

− − +

b) ( )( )

c) ( )( ) ( ) ( )

1 2 2 3

t t t t

+ +−

+ + + +

w ww+ −

− +−

2 1x +,

( )( )

1 3t t+ +,

( )( ) 2

2 2 4w w w=

− + −

Created by T. Madas

Question 5

b) ( )( ) ( )( )

3 4 3 2 1

y y y y

− −−

− + − −

2 5 2 7 5

t t t+ −

− − +

d) ( ) ( )

ww w+ −

3x −,

( )( )

+ −,

− −

Created by T. Madas

Question 6

a) ( )

4 3 14 8

−−

b) 2 2

y y y−

− − +

− −

+ +−

−− −

3 2x +,

( )( )

− +,

t −, 1−

Created by T. Madas

Question 7

Prove that

3 4 2 12 4

x x x x

− − +− ≡

+ −−

b) ( )2

3 45 11 9 2 3

2 53 10

xx x x

x xx x

−− + −− ≡

− ++ −

c) ( )2

2 15 10 15 3 3

12 39 36

xx x x

x xx x

+− − +− ≡

− +− −

d) ( )( )

( )( )2

1 3 1 39

x x x xx

+++ ≡

+ + + −−

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Question 8

Prove that

a) ( )

2 2 32 3 2 9

2 2 12 3 2

x xx x

−+ +− ≡

+ −+ −

2 3 6 3

12 6 2

xx x x x

+ +− ≡

+− − − −

8 2 31

4 22 8

x xx x

− −+ − ≡

+ −+ −

1 5 3 1

x xx x

+ −− + ≡

− ++ −

Created by T. Madas

Question 9

Prove that

2 11 3

1 43 4

x xx x

−+ ≡

− ++ −

3 2 1 3 1

xx x x

− − −− ≡

c) ( ) ( )( )

4 1 3 32

2 1 2 1 2 1 2 1

x x x x

−− − ≡

− − − −

x xx x− + ≡

− +− −

Created by T. Madas

Question 10

Prove that

2 13 2

x xx x

+− ≡

− −− +

2 1 13 2

x x xx x

−− − ≡

+ + ++ +

c) ( )

( )( )2 2

2 14 1 5

2 3 42 6 2 8

x xx x x x

+−− ≡

+ +− − + −

12 3 3 3

1 32 3

x xx x

+ +− + ≡

− ++ −

Created by T. Madas

ALGEBRAIC

DIVISIONS

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Question 1

Find the values of the constants in the each of the following identities.

a) 3 2

22 7 1

x x x dax bx c

− + −≡ + + +

b) 3 2

22 3 2

x x x DAx Bx C

− − +≡ + + +

− −

26 4 8

x x SPx Qx R

− +≡ + + +

( ) ( ), , , 1, 3,10, 11a b c d = − − , ( ) ( ), , , 2,1,1,4A B C D = , ( ) ( ), , , 6,12,20, 32P Q R S = −

Created by T. Madas

Question 2

a) 3 2

22 3 1

x x x dax bx c

+ − +≡ + + +

− −

b) 3 2

22 3 2

x x x DAx Bx C

− + −≡ + + +

c) 3 2

22 3 4

x x x SPx Qx R

− + −≡ + + +

− −

( ) ( ), , , 1,4,5,11a b c d = , ( ) ( ), , , 2, 3,6, 8A B C D = − − , ( ) ( ), , , 2, 1,0, 4P Q R S = − −

Created by T. Madas

Question 3

a) 3 2

23 20 31

x x x dax bx c

− − +≡ + + +

b) 3 2

24 7 30 21

x x x DAx Bx C

− − −≡ + + +

− −

22 7 2

x x SPx Qx R

− +≡ + + +

( ) ( ), , , 3, 10,10,1a b c d = − , ( ) ( ), , , 4,9,6,3A B C D = , ( ) ( ), , , 2, 2, 5,7P Q R S = − −

Created by T. Madas

Question 4

a) 3 2

x x x cx dax b

− + − +≡ + +

b) 3 2

x x x Cx DAx B

− − + +≡ + +

− −

x x Rx SPx Q

− + +≡ + +

− −

( ) ( ), , , 1, 2,1,1a b c d = − , ( ) ( ), , , 3, 1,2,0A B C D = − , ( ) ( ), , , 5,0,1,5P Q R S =

Created by T. Madas

Question 5

a) 3 2

x x x cx dax b

x x x x

− − + +≡ + +

+ − + −

x x Cx DAx B

+ − +≡ + +

x x Rx SPx Q

x x x x

+ − +≡ + +

− −

( ) ( ), , , 3, 4,5, 3a b c d = − − , ( ) ( ), , , 2,0, 1, 2A B C D = − − , ( ) ( ), , , 2,1,2, 4P Q R S = −

Created by T. Madas

Question 6

a) 3 2

x x x cax b

x dx x

+ − +≡ + +

++ −

b) 3 2

2 3 10 3

x x x CAx B

x Dx x

+ − −≡ + +

++ −

c) 3 2

x x x RPx Q

x Sx x

− − − +≡ + +

( ) ( ), , , 2, 1,1,2a b c d = − , ( ) ( ), , , 2,1,1, 2A B C D = − , ( ) ( ), , , 1, 2,4,1P Q R S = − −

Created by T. Madas

MULTIPLICATION

DIVISION

ALGEBRAIC

FRACTIONS

Created by T. Madas

Question 1

fraction in its simplest form:

a) 3 2

4 2 5 3

+ −×

− − −

b) 2 2

18 8 3 9

12 3 11 6

− +×

c) 3 2

1 4 25

2 7 5 4 1

t t t t

+ −×

+ + − +

( )( )2

x x+ +,

2 5t −

Question 2

x x x x÷

+ − −

b) ( )

8 15 9

− + −÷

2 5x x+,

Created by T. Madas

Question 3

−−

++ −

c) ( )( ) ( )2 1 2 5 4 2 1

+ − + −

43x−

, 54x+

, 24 4 3x x− −

Created by T. Madas

PARTIAL

FRACTIONS

Created by T. Madas

Question 1

Express each of the following into partial fractions.

a) ( )( )

x x− +

b) ( ) ( )

c) ( )( )

d) ( )( )

e) ( )( )

2 41 2x x− +

+ , 523 2y y− +

+ , 513 4t t− +

− , 3 22 6w w+ −

− , 7172 zz −+

Created by T. Madas

Question 2

a) ( )( )

b) ( )( )

c) ( )( )

d) ( )( )

1 41 3x x+ −

+ , 4 15 2y y− +

+ , 2 12 1t t− +

− , 342 1 4w w+ +

Created by T. Madas

Question 3

a) ( )( )

b) ( )( )

2 1 2y y+ −

c) ( ) ( )

2 1 2 1

d) ( )( )

7 154 xx −+

+ , 2 12 1 2y y+ −

− , 2 12 1 2 1t t+ −

− , 6 551 ww −+

Created by T. Madas

Question 4

− −

8 18 9

323 2x x− −

− , 6 14 1y y− +

+ , 5 22 3 2t t− +

− , 2 43 4 3 2w w− −

Created by T. Madas

Question 5

a) ( ) ( )

b) ( )( )

2 5 12

d) ( )( )( )

26 12 9

1 2 1 2

+ − −

6 51 2x x+ +

− , 4 15 3y y+ −

− , 31 22 3tt t− +

− + , 31 22 2 1 1w w w− − +

Created by T. Madas

Question 6

a) ( )( )

b) ( )( )( )

1 3 2 1y y y+ + +

c) ( )( ) ( )

24 5 3

+ − −

d) ( )( ) ( )

212 4 3

2 1 2 1 2 3

+ − −

1 1 22 1 1x x x− +

+ − , 5 813 1 2 1y y y+ + +

− + , 32 11 1 2t t t+ − −

− + , 31 12 1 2 1 2 3w w w+ − −

Created by T. Madas

Question 7

a) ( )( )

− −

b) ( ) ( )

c) ( ) ( )

3 12 7

− + +

d) ( )( )

3 10 11

− + −

− −

3 1 22 1 1x x x− − −

− + , ( )

22 yy +−+ ,

1 4 23 1 1t t t− + +

− + , ( )

21 ww −−−

Created by T. Madas

Question 8

a) ( )( )

− −

b) ( )

3 17 4

c) ( )( )

t t− −

d) ( )( )

w w+ −

6 2 13 2 1 1x x x− − −

− − , 21 4 1

4y yy ++ − ,

34 12 11t tt− −−

− − , ( )

1 2 11w ww− +−+ +

Created by T. Madas

Question 9

a) ( )( )

b) ( ) ( )

c) ( )

d) ( )( )

− −

1 2 12 2 1 2 1x x x+ − −

+ − , ( )

+ , 26 311 tt t−

− − , ( )

3 11w ww− −−− −

Created by T. Madas

Question 10

a) ( )( )

b) ( ) ( )

5 19 13

− −

c) ( ) ( )

15 13 4

− −

d) ( )( )

− −

6 3 82 1 1 1x x x− + +

− + , ( )

−−

+ , ( )

23 1 2

− −−

+ + , ( )

26 1 2

3 2 11w ww− −−− −

Created by T. Madas

Question 11

a) ( )( )

22 8 5

− −

b) ( )( )

24 5 15

− −

c) ( )( )

3 22 7 6 3

− + −

− −

d) 3 2

− − +

2x x− −

+ − , 321 2

4x x+ −

+ − , 321 2

2 1x x

x− −

− + − , 1 41 1

x− +

− + −

Created by T. Madas

Question 12

a) ( ) ( )

c) ( )( )

d) ( )( )

23 31 45

− −

( ) ( )921

2 5 1 2 12

x x+ ++ − , 4 4

y y− ++ − , 34

2 3 21

t t− ++ − , 51

ww −−+ −

Created by T. Madas

Question 13

a) ( )( )2

b) ( )( )2

1 2y y+ −

c) ( )( )

d) ( )( )

x x−

− +,

t t−

− +,

+ +−

Created by T. Madas

Question 14

a) ( )( )2

1 1x x+ −

b) ( )( )

c) ( )( )

2 1 6 1

− +,

9 9y y+

6 1 2 1

w w w− +

− + +

Created by T. Madas

Question 15

a) ( )( )

b) ( )( )

c) ( )( )

2 1 6 1

( )2 2

3 2x x−

+ −,

9 9y y+

6 1 2 1

w w w− +

− + +

Created by T. Madas

Question 16

a) ( )( )

b) ( )( )

c) ( )( )

2 1 6 1

3 2 32

++ −

− +,

9 9y y+

6 1 2 1

w w w− +

− + +

algebraic fractions - MadAsMaths · Created by T. Madas Created by T. Madas Question 2 Simplify the...

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