To find the partial fraction decomposition of the rational expression [tex]\(\frac{-x^2 - x + 16}{x^3 + 8x^2 + 16x}\)[/tex], follow these steps:
1. Factorize the denominator:
The denominator [tex]\(x^3 + 8x^2 + 16x\)[/tex] can be factored as follows:
[tex]\[
x^3 + 8x^2 + 16x = x(x^2 + 8x + 16) = x(x+4)^2
\][/tex]
2. Express the rational function as a sum of partial fractions:
Given the factorized form of the denominator, we can rewrite the rational expression as a sum of partial fractions. The general form for the partial fraction decomposition is:
[tex]\[
\frac{-x^2 - x + 16}{x(x+4)^2} = \frac{A}{x} + \frac{B}{x+4} + \frac{C}{(x+4)^2}
\][/tex]
3. Set up the equation by combining the fractions on the right side:
[tex]\[
\frac{A}{x} + \frac{B}{x+4} + \frac{C}{(x+4)^2} = \frac{A(x+4)^2 + Bx(x+4) + Cx}{x(x+4)^2}
\][/tex]
Now, we want this to match the original numerator [tex]\(-x^2 - x + 16\)[/tex]:
[tex]\[
-x^2 - x + 16 = A(x+4)^2 + Bx(x+4) + Cx
\][/tex]
4. Expand and match coefficients:
Expand the right-hand side:
[tex]\[
A(x+4)^2 = A(x^2 + 8x + 16)
\][/tex]
[tex]\[
Bx(x+4) = Bx^2 + 4Bx
\][/tex]
[tex]\[
Cx = Cx
\][/tex]
Combining these, we get:
[tex]\[
A(x^2 + 8x + 16) + B(x^2 + 4x) + Cx = Ax^2 + 8Ax + 16A + Bx^2 + 4Bx + Cx
\][/tex]
[tex]\[
= (A + B)x^2 + (8A + 4B + C)x + 16A
\][/tex]
Now we will match the coefficients with the left-hand side [tex]\(-x^2 - x + 16\)[/tex]:
Comparing coefficients, we get:
[tex]\[
A + B = -1
\][/tex]
[tex]\[
8A + 4B + C = -1
\][/tex]
[tex]\[
16A = 16
\][/tex]
5. Solve the system of equations:
From [tex]\(16A = 16\)[/tex]:
[tex]\[
A = 1
\][/tex]
Substituting [tex]\(A = 1\)[/tex] into [tex]\(A + B = -1\)[/tex]:
[tex]\[
1 + B = -1 \implies B = -2
\][/tex]
Substituting [tex]\(A = 1\)[/tex] and [tex]\(B = -2\)[/tex] into [tex]\(8A + 4B + C = -1\)[/tex]:
[tex]\[
8(1) + 4(-2) + C = -1
\][/tex]
[tex]\[
8 - 8 + C = -1 \implies C = -1
\][/tex]
6. Write the partial fraction decomposition:
The partial fraction decomposition is therefore:
[tex]\[
\frac{-x^2 - x + 16}{x(x+4)^2} = \frac{1}{x} + \frac{-2}{x+4} + \frac{-1}{(x+4)^2}
\][/tex]
Hence, the correct option is:
[tex]\[
\boxed{\frac{1}{x}+\frac{-2}{x+4}+\frac{-1}{(x+4)^2}}
\][/tex]
