To find the form of the partial fraction decomposition for the rational expression [tex]\(\frac{9x^3 + x^2}{(x^2 + 3)^2}\)[/tex], we need to consider the nature of the denominator. The denominator [tex]\((x^2 + 3)^2\)[/tex] is a repeated quadratic factor. When decomposing a rational expression with a repeated quadratic factor, we use the following general approach:
1. Identify the basic quadratic factor: In this case, the basic quadratic factor is [tex]\(x^2 + 3\)[/tex].
2. Account for repeated factors: Since the denominator is [tex]\((x^2 + 3)^2\)[/tex], we need separate terms in the decomposition for both [tex]\(x^2 + 3\)[/tex] and [tex]\((x^2 + 3)^2\)[/tex].
3. Set up the partial fraction form:
- For the basic quadratic factor [tex]\((x^2 + 3)\)[/tex], the numerator should be of one degree lower than the quadratic factor. Therefore, it should be in the form [tex]\(Ax + B\)[/tex].
- For the repeated factor [tex]\((x^2 + 3)^2\)[/tex], similarly, the numerator should be of one degree lower than the quadratic factor. Consequently, it should also be in the form [tex]\(Cx + D\)[/tex].
Given this structure, the partial fraction decomposition of the expression [tex]\(\frac{9x^3 + x^2}{(x^2 + 3)^2}\)[/tex] will be:
[tex]\[
\frac{Ax + B}{x^2 + 3} + \frac{Cx + D}{(x^2 + 3)^2}
\][/tex]
Therefore, the form of the partial decomposition is correctly described by:
B. [tex]\(\frac{A x + B}{x^2 + 3} + \frac{C x + D}{(x^2 + 3)^2}\)[/tex]
