Understanding Engineering Mathematics

(vii)

x

x^2 − 1

+

1

x− 1

(viii) x−

1

x+ 1

+

2

x+ 2

(ix)

3

x^2 + 1

+

2

x^2 + 2

(x) 2x− 1 +

2

x− 1

−

3

x+ 2

B.Put over a common denominator:

(i)

2

x− 4

+

3

x− 1

(ii)

4

x− 4

−

2

x− 4

(iii)

x− 1

(x− 2 )^2

+

3

x− 1

(iv)

2 x− 1

x^2 + 1

−

4

x− 2

(v) 1+

3

x− 2

2.3.9 Division and the remainder theorem

➤➤

39 60

➤

A.Perform the following divisions, whenever permissible:

(i)

x^2

x^2

(ii)

x^2 − 2 x+ 2

(x− 1 )^2

(iii)

x^4 − 2 x^3 + 4 x− 1

x− 3

(iv)

x^3 + 1

x+ 1

(v)

2 x^2 − 3 x+ 4

x^2 − 1

(vi)

x^4 −y^4

x^2 +y^2

B.Find the remainders when the following polynomials are divided by:

(a) x−1(b)x+2(c)x

(i) 3x^3 + 2 x− 1 (ii) x^5 − 2 x^2 + 2 x− 1

(iii) x^4 −x^2 (iv) 2x^7 − 3 x^5 + 4 x^3 − 2 x^2 + 1

2.3.10 Partial fractions

➤➤

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➤

A.For Q2.3.8A, B check your results by reversing the operation and resolving your answer
into partial fractions. Usually, the answers should of course be what you started with
in those questions. There are however a couple of cases where this is not so – explain
these cases.
B.Split into partial fractions:

(i)

x− 1

(x+ 2 )(x− 3 )

(ii)

4

x^2 − 1

(iii)

x+ 1

x(x− 3 )

(iv)

2 x+ 1

(x− 1 )^2 (x+ 2 )

(v)

3

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

(vi)

5 x− 4

(x^2 + 4 )(x− 2 )^2

(vii)

x+ 1

(x− 1 )(x+ 3 )(x− 4 )