For the given functions $f(x)$ and $g(x)$, find $f(x) \div g(x)$.
-$f(x)=\frac{x^{2}+8 x+15}{x^{2}+12 x+35}, g(x)=\frac{x^{2}+3 x}{x^{2}-2 x-63}$

A) $\frac{x}{x^{2}+12 x+35}$

B) $\frac{x-9}{x^{2}+7 x}$

C) $x-9$

D) $\frac{x-9}{x}$

Question

For the given functions $f(x)$ and $g(x)$, find $f(x) \div g(x)$.
-$f(x)=\frac{x^{2}+8 x+15}{x^{2}+12 x+35}, g(x)=\frac{x^{2}+3 x}{x^{2}-2 x-63}$

A) $\frac{x}{x^{2}+12 x+35}$

B) $\frac{x-9}{x^{2}+7 x}$

C) $x-9$

D) $\frac{x-9}{x}$

Quizplus · Accepted Answer

To find $f(x) \div g(x)$, we first factorize the numerators and denominators of $f(x)$ and $g(x)$ if possible. $f(x)=\frac{x^{2}+8 x+15}{x^{2}+12 x+35} = \frac{(x+5)(x+3)}{(x+7)(x+5)}$$g(x)=\frac{x^{2}+3 x}{x^{2}-2 x-63} = \frac{x(x+3)}{(x-9)(x+7)}$Now, $f(x) \div g(x) = \frac{(x+5)(x+3)}{(x+7)(x+5)} \div \frac{x(x+3)}{(x-9)(x+7)}$This simplifies to $\frac{(x+5)(x+3)}{(x+7)(x+5)} \cdot \frac{(x-9)(x+7)}{x(x+3)}$Canceling common factors, we get $\frac{x-9}{x}$.Therefore, the correct answer is D.

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