Partial fractions, need help ?

Question

Q. express in partial fractions

1. x^2

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(x-1)(x+2)

2. x^3

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(x^2 -1)

i tried these but i am not sure if i am doing them correctly

thanks in advance

Answer 1

1.]     x^2/[(x-1)(x+2) which is in improper form since the degree of the numerator is not less than the degree of the denominator. We must first divide the denominator into the numerator to get it into the proper form:

(x-1)(x+2) = x^2+x-2

            

x^2+x-2 divided into x^2 using polynomial division gives the following:

x^2/[(x-1)(x+2) = 1 + (2-x)/[(x-1)(x+2)]

Breaking (2-x)/[(x-1)(x+2)] into partial fractions:

(2-x)/[(x-1)(x+2)] = A/(x-1) + B/(x+2)

2 - x = A(x + 2) + B(x - 1)

so -1 = A + B and 2 = 2A - B

A = 1/3 and B = -4/3

Therefore:

x^2/[(x-1)(x+2) = 1 + 1/[3(x-1)] - 4/[3(x+2)]

2.] APPROACH IS SIMILAR TO 1.]