Understanding Engineering Mathematics

2.3.8 Algebra of rational functions

A. (i)

−x− 8

(x+ 2 )(x− 1 )

(ii)

−x− 10

(x+ 3 )(x− 4 )

(iii)

−x− 13

(x+ 3 )(x− 2 )

(iv)

5 x− 12

(x− 2 )(x− 3 )

(v)

3 x^2 − 12 x+ 11

(x− 1 )(x− 2 )(x− 3 )

(vi)

− 4 x^2 − 17 x− 27

(x+ 2 )(x+ 3 )(x− 1 )

(vii)

2 x+ 1

x^2 − 1

(viii)

x^3 + 3 x^2 + 3 x

(x+ 1 )(x+ 2 )

(ix)

5 x^2 + 8

(x^2 + 1 )(x^2 + 2 )

(x)

2 x^3 +x^2 − 6 x+ 9

(x− 1 )(x+ 2 )

B. (i)

5 x− 14

(x− 1 )(x+ 2 )

(ii)

2

x− 4

(iii)

4 x^2 − 14 x+ 13

(x− 1 )(x− 2 )^2

(iv)

− 2 x^2 − 5 x− 2

(x^2 + 1 )(x− 2 )

(v)

x+ 1

x− 2

2.3.9 Division and the remainder theorem

A. (i) 1, x = 0 (ii) 1+

1

(x− 1 )^2

x   = 1

(iii) x^3 +x^2 + 3 x+ 13 +

38

x− 3

x   = 3

(iv) x^2 −x+ 1 x    =−1(v)2−

3 (x− 2 )

x^2 − 1

x   =± 1

(vi) x^2 −y^2

B. (i) (a) 4 (b)− 29 (c)− 1

(ii) (a) 0 (b)− 45 (c)− 1

(iii) (a) 0 (b) 28 (c) 0

(iv) (a) 0 (b)− 199 (c) 1

2.3.10 Partial fractions

A.For answers see Q2.3.8A, B.

B. (i)

3

5 (x+ 2 )

+

2

5 (x− 3 )

(ii)

2

x− 1

−

2

x+ 1

(iii)

4

3 (x− 3 )

−

1

3 x

(iv)

1

(x− 1 )^2

−

1

x− 1

−

3

x+ 2

(v)

3

2 (x+ 1 )

−

3

2

(x− 1 )

x^2 + 1

(vi)

1

4 (x− 2 )

+

3

4 (x− 2 )^2

+

1

4

(x+ 5 )

x^2 + 4

(vii) −

1

6 (x− 1 )

−

1

14 (x+ 3 )

+

5

28 (x− 4 )