partial fractions - solutions.pdf

4
AQA Core 4 Algebra 1 of 4 18/03/13 © MEI Section 2: Partial fractions Solutions to Exercise 1.  i) 1  1 1 1 1 xx xx  x x  Putting 1  Putting 1 1   1 1 1  1 1 x x x  ii) 14  2 1 3 2 1 3 14 3 2 1 x x x x x x x x  Putting 42 7 6  Putting 2 2 2   14 2 6  2 1 3 2 1 3 x x x x x  2.   3 2  1 2 2 3 1 2 2 3 3 2 2 2 3 1 2 3 1 2 x C x x x x x x x x x x x Cx x  Putting  1 3 5 15  Putting 4 3 4 3  Putting 5 5 2 2 4 2 C C   3 2 1 4 2  1 2 2 3 3 1 3 2 2 3 x x x x x x x  3.  i) 7  1 2 1 2 7 2 1 x x x x x x x x  Putting 6 3 2  Putting 93 3   7 3 2  1 2 2 1 x x x x x  ii) 2 1 1 4 2 2 2 2 1 2 2 x x x x x x x x x x  Putting  3 4 3 4  Putting  1 4 1 4  

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8/17/2019 Partial Fractions - Solutions.pdf

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AQA Core 4 Algebra

1 of 4  18/03/13 © MEI 

Section 2: Partial fractions

Solutions to Exercise

1.  i)

1

  1 1

1 1

x x x x

  x x

 

Putting 1

 

Putting 1 1

 

1 1 1

  1 1

x x x

 

ii)

14

 2 1 3 2 1 3

14 3 2 1

x

x x x x

x x x

 

Putting 42 7 6

 

Putting 2 2

14 2 6

 2 1 3 2 1 3

x

x x x x

 

2. 

3 2

  1 2 2 3 1 2 2 3

3 2 2 2 3 1 2 3 1 2

x C

x x x x x x

x x x x x C x x

 

Putting  1

3

5 15

 

Putting 4

3

4 3

 

Putting 5 5

2 2 4

2

C C

 

3 2 1 4 2

  1 2 2 3 3 1 3 2 2 3

x

x x x x x x

 

3.  i)

7

  1 2 1 2

7 2 1

x

x x x x

x x x

 

Putting 6 3 2

 

Putting

9 3 3

 

7 3 2

  1 2 2 1

x

x x x x

 

ii)

2

1 1

4 2 2 2 2

1 2 2

x x

x x x x x

x x x

 

Putting  3

4

3 4

 

Putting  1

4

1 4

 

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  AQA C4 Algebra 2 Exercise solutions

2 of 4  18/03/13 © MEI 

1 3 1

4 4 2 4 2

x

x x x

 

iii)

2

2

9

  1 3 2 1 1 3 2 1

9 3 2 1 1 2 1 1 3

x x

  C

x x x x x x

x x x x x x C x x

 

Putting 8 4 2

 

Putting 84 28 3

 

Putting 7

2 4 4

1C C  

2

9

2 3 1

  1 3 2 1 1 3 2 1

x x

x x x x x x

 

4.  i)

2 2

2

  1 1 1

2 1

x

x x x

x x

Putting 2

 

Equating coefficients of x

 

2

2 2 2

  1 1 1

x

x x x

 

ii)

2 2

3 2

 2 3 2 3 2 3

3 2 2 3

x

x x x

x x

Putting 3

4  

Equating coefficients of x 3 1

 

2

3 2 4 1

 2 3 2 3 2 3

x

x x x

 

5.  i)

2 2

2

4

 1 3 1 1 3 1 1

4 1 1 3 1 1 3

C

x x x x x

  x x x C x

 

Putting 3 9

Putting 4 2 2

C C  

Equating coefficients of x²

0 3

3 9

3

 

2

9

3 2

 1 3 1 1 3 1 1x x x x

 

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  AQA C4 Algebra 2 Exercise solutions

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ii)

2 2

2

13 7

  2 3 2 2 3

13 7 2 3 3 2

x C

x x x x x

x x x x C x

 

Putting 1 1

 

Putting 8

C

 

Equating coefficients of x² 8

C

2

13 7 8 1 8

  2 3 2 2 3

x

x x x x x

 

6.  i) The degree of the numerator and denominator are both 2, so there

will be a constant term.

2

2

2 6 7

  2 2 1 2 2 1

2 6 7 2 2 1 2 1 2

x x C

 

x x x x

x x x x x C x

 

Putting 3 3 1

 

Putting 1 3

2 2 2

3C C  

Equating coefficients of x² 2 1

 

2

2 6 7 1 3

1

  2 2 1 2 2 1

x x

x x x x

 

ii) The degree of the numerator is 3 and the degree of the denominator is 2, so

there will be a linear expression.

3 2

3 2 2

2 4 1

  1 1

2 4 1 1 1 1

x x x C D

 x

x x x x

x x x x x x x C x Dx

 

Putting 1

C

 

Putting 2 2

D D

 

Equating coefficients of x³

 

Equating coefficients of x² 3 

3 2

2 4 1 1 2

2 3

  1 1

x x x

x

x x x x

 

iii) The degree of the numerator and denominator are both 3, so there will be a

constant term.

3 2

2 2

3 2 2

2

3 9 2

  1 4 1 1 4

3 9 2 1 4 1 4

4 1

x x x

  C D

 

x x x x x

x x x x x x x

C x D x

 

Putting

15 5 3

C C  

Putting

50 25 2

D D

 

Equating coefficients of x³  

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  AQA C4 Algebra 2 Exercise solutions

4 of 4  18/03/13 © MEI 

Equating constant terms

2 4 4 4

2 4 4 12 2

3

  C D

 

3 2

2 2

3 9 2

3 3 2

1

  1 4 1 1 4

x x x

x x x x x