Write the form of the partial fraction decomposition of the rational expression. It is not necessary to solve for the constants.

[tex]\[ \frac{9 x^3+x^2}{\left(x^2+3\right)^2} \][/tex]

What is the form of the partial fraction decomposition of the given expression?

A. [tex]\(\frac{A}{x^2+3} + \frac{B}{\left(x^2+3\right)^2}\)[/tex]

B. [tex]\(\frac{A x+B}{x^2+3} + \frac{C x+D}{\left(x^2+3\right)^2}\)[/tex]

C. [tex]\(\frac{A x+B}{x+3} + \frac{C x+D}{x^2+3}\)[/tex]

D. [tex]\(\frac{A x+B}{x^2+3}\)[/tex]

E. [tex]\(\frac{A x+B}{x+3} + \frac{C x+D}{x^2+3} + \frac{E x+F}{\left(x^2+3\right)^2}\)[/tex]

F. [tex]\(\frac{A x+B}{(x+3)^2}\)[/tex]



Answer :

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]