simplify the complex rational expression. $\frac{\frac{7}{(x + h)^2}-\frac{7}{x^2}}{h}$ select the correct…

simplify the complex rational expression. $\frac{\frac{7}{(x + h)^2}-\frac{7}{x^2}}{h}$ select the correct choice below and fill in any answer box(es) within your choice. (simplify your answer. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) a. $\frac{\frac{7}{(x + h)^2}-\frac{7}{x^2}}{h}=square,x\neqsquare,h\neqsquare$ b. $\frac{\frac{7}{(x + h)^2}-\frac{7}{x^2}}{h}=square,x\neqsquare$, and no numbers must be excluded for h. c. $\frac{\frac{7}{(x + h)^2}-\frac{7}{x^2}}{h}=square,h\neqsquare$, and no numbers must be excluded for x. d. $\frac{\frac{7}{(x + h)^2}-\frac{7}{x^2}}{h}=square$, no numbers must be excluded.

simplify the complex rational expression. $\frac{\frac{7}{(x + h)^2}-\frac{7}{x^2}}{h}$ select the correct choice below and fill in any answer box(es) within your choice. (simplify your answer. use integers or fractions for any numbers in the expression. use a comma to separate answers as needed.) a. $\frac{\frac{7}{(x + h)^2}-\frac{7}{x^2}}{h}=square,x\neqsquare,h\neqsquare$ b. $\frac{\frac{7}{(x + h)^2}-\frac{7}{x^2}}{h}=square,x\neqsquare$, and no numbers must be excluded for h. c. $\frac{\frac{7}{(x + h)^2}-\frac{7}{x^2}}{h}=square,h\neqsquare$, and no numbers must be excluded for x. d. $\frac{\frac{7}{(x + h)^2}-\frac{7}{x^2}}{h}=square$, no numbers must be excluded.

Answer

Explanation:

Step1: Find common denominator for numerator

First, find a common - denominator for $\frac{7}{(x + h)^2}-\frac{7}{x^2}$. The common denominator is $x^{2}(x + h)^{2}$. So, $\frac{7}{(x + h)^2}-\frac{7}{x^2}=\frac{7x^{2}-7(x + h)^{2}}{x^{2}(x + h)^{2}}$.

Step2: Expand the numerator

Expand $7x^{2}-7(x + h)^{2}$: [ \begin{align*} 7x^{2}-7(x + h)^{2}&=7x^{2}-7(x^{2}+2xh+h^{2})\ &=7x^{2}-7x^{2}-14xh - 7h^{2}\ &=-14xh-7h^{2} \end{align*} ] So, $\frac{7}{(x + h)^2}-\frac{7}{x^2}=\frac{-14xh - 7h^{2}}{x^{2}(x + h)^{2}}$.

Step3: Divide by h

We have $\frac{\frac{7}{(x + h)^2}-\frac{7}{x^2}}{h}=\frac{\frac{-14xh - 7h^{2}}{x^{2}(x + h)^{2}}}{h}=\frac{-14xh - 7h^{2}}{x^{2}(x + h)^{2}}\cdot\frac{1}{h}$. Factor out $-7h$ from the numerator: $\frac{-7h(2x + h)}{x^{2}(x + h)^{2}}\cdot\frac{1}{h}$. Cancel out the common factor $h$ (where $h\neq0$), we get $\frac{-7(2x + h)}{x^{2}(x + h)^{2}}=\frac{-14x-7h}{x^{2}(x + h)^{2}}$. Since we divided by $h$, $h\neq0$. Also, the original expression has denominators $(x + h)^2$ and $x^2$, so $x\neq0$ and $x\neq - h$.

Answer:

A. $\frac{\frac{7}{(x + h)^2}-\frac{7}{x^2}}{h}=\frac{-14x - 7h}{x^{2}(x + h)^{2}},x\neq0,x\neq - h,h\neq0$