Find the partial fraction decomposition for the rational expression.
-$$\frac { x ^ { 2 } } { x ^ { 2 } + 6 x + 9 }$$
A) $\frac { 6 } { x + 3 } + \frac { - 6 } { ( x + 3 ) ^ { 2 } }$

B) $\frac { - 6 } { x + 3 } + \frac { 9 } { ( x + 3 ) ^ { 2 } }$

C) $1 + \frac { 6 } { x + 3 } + \frac { - 9 } { ( x + 31 ) ^ { 2 } }$

D) $1 + \frac { - 6 } { x + 3 } + \frac { 9 } { ( x + 3 ) ^ { 2 } }$

Question

Find the partial fraction decomposition for the rational expression.
-$$\frac { x ^ { 2 } } { x ^ { 2 } + 6 x + 9 }$$
A) $\frac { 6 } { x + 3 } + \frac { - 6 } { ( x + 3 ) ^ { 2 } }$

B) $\frac { - 6 } { x + 3 } + \frac { 9 } { ( x + 3 ) ^ { 2 } }$

C) $1 + \frac { 6 } { x + 3 } + \frac { - 9 } { ( x + 31 ) ^ { 2 } }$

D) $1 + \frac { - 6 } { x + 3 } + \frac { 9 } { ( x + 3 ) ^ { 2 } }$

Quizplus · Accepted Answer

The denominator $x^2 + 6x + 9$ can be factored as $(x + 3)^2$. The numerator $x^2$ is already in its simplest form. The partial fraction decomposition of a rational expression where the denominator is a square of a binomial (like $(x + 3)^2$) typically involves terms of the form $A/(x + 3)$ and $B/(x + 3)^2$, plus possibly a polynomial term if the degree of the numerator is equal to or greater than the degree of the denominator. In this case, since the degrees of the numerator and denominator are the same, we expect a constant term as well. The correct decomposition must then start with a constant term (which is $1$ here, because $x^2/(x^2 + 6x + 9)$ simplifies to $1$ when $x^2$ is subtracted from both numerator and denominator), followed by terms that correctly decompose the remainder. Option D correctly represents this decomposition, with the constant term $1$ and the terms $\frac{-6}{x+3}$ and $\frac{9}{(x+3)^2}$ accounting for the partial fraction decomposition of the remainder.

Find the Partial Fraction Decomposition for the Rational Expression x2x2+6x+9\frac { x ^ { 2 } } { x ^ { 2 } + 6 x + 9 }x2+6x+9x2​

Find the Partial Fraction Decomposition for the Rational Expression $\frac { x ^ { 2 } } { x ^ { 2 } + 6 x + 9 }$