Question

Find the partial fraction decomposition of the rational expression with repeated factors, (-x²-x+16)/(x³+8x²+16x)

Answer 1

The student is requesting the partial fraction decomposition of a rational expression with repeated linear factors, which requires factoring the denominator and setting up a decomposition that aligns with the form of the original denominator's factors.

Step-by-step explanation:

The question is asking for the partial fraction decomposition of the given rational expression with repeated linear factors in the denominator. To find the partial fraction decomposition of
`\((-x^2-x+16)/(x^3+8x^2+16x)\),`, recognizing that it represents a polynomial with repeated linear factors. The complete factorization of
`\(x^3+8x^2+16x\) is \(x(x+4)^2\).`ition will take the form:

`\[(-x^2-x+16)/(x(x+4)^2) = (A)/(x) + (B)/(x+4) + (C)/((x+4)^2)\]`

Next, you would multiply both sides by the denominator to remove the fraction and equate the coefficients of corresponding powers of
`\(x\)` sides to find the values of
`\(A\), \(B\), and \(C\)` manipulation and possibly solving a system of equations.