Question

Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants.

Answer 1

Factorise and split the denominator:

1/(x^3 + 8x) = 1/(x(x^2+8)) = (1/x) * (1/(x^2+8))

now say

(a/x) + ((bx+c)/(x^2+8)) = (x+16)/(x^3+8x)

so

(x^2+8)a + (bx+c)x = x + 16

a must be 2 to get the + 16

2x^2 + 16 + bx^2 + cx = x + 16

b must be -2 to get rid of the 2x^2

c must be 1 to get the x

`(2)/(x) + (1-2x)/(x^2+8)`