How would I perform partial fraction decomposition on the equation: 17/[(x-1)(x^2+16)]?

17 / [(x-1)(x²+16)] = A/(x-1) + (Bx+C)/(x²+16)

{Note: The first degree linear numerator is necessary when the factor is an "irreducible quadratic factor"}

Multiply the equation by (x-1)(x²+16):

17 = A(x²+16) + (Bx+C)(x-1)

When x = 1, we see 17 = 17A --> A = 1

Since there are no other convenient values of x, we can use matching like terms on either side of the equation to find B and C:

On the LHS, there is Ax² +Bx², which must equal 0x² on the RHS. Therefore, B = -1, since A = 1.

On the LHS, the constant terms are 16A - C, which must equal 17 on the RHS. Since A = 1, then C = -1.

So partial fraction decomposition is: 1/(x-1) + (-x-1)/(x²+16)