How do you write the partial fraction decomposition of the rational expression 10/[(x-1)(x^2+9)] ?

1 Answer
Jan 27, 2018

1/(x-1)+(-x-1)/(x^2+9)

Explanation:

We can rewrite the function as:

10/((x-1)(x^2+9))=A/(x-1)+(Bx+C)/(x^2+9)

and now we need to work out A and B.

Multiply both sides by (x-1)(x^2+9) to get:

10 = A(x^2+9)+(Bx+C)(x-1)

Let x=1 to cancel the second term and get:

10=A(1^2+9)=10A implies A=1

Now let x=0 to cancel the B, we get:

10 = 1*(0^2+9)+C(0-1)

implies 10=9-C implies C = -1

Finally, choose any other value of x to get B, say x=2

implies 10 =1*(2^2+9)+(B(2)-1)(2-1)

implies 10 = 13+2B-1

implies B = -1

Now putting our values in:

10/((x-1)(x^2+9))=1/(x-1)+(-x-1)/(x^2+9)