# Multiply the following rational expressions and simplify the result....

## Question:

Multiply the following rational expressions and simplify the result.

{eq}\frac{5y^2}{5y^2+30y-35}.\frac{2y^3+30y^2+112y}{8y+44} {/eq}

Which values of y make the resulting expression undefined?

A. y = -7

B. y = -1

C. y = {eq}-\frac{11}{2} {/eq}

D. y = 1

E. y = 0

F. None of the above

## Rational Expression:

A rational expression is an expression that is given in the form of {eq}\dfrac{p(x)}{q(x)} {/eq} form, where the numerator {eq}p(x) {/eq} or the denominator {eq}q(x) {/eq} or both are polynomials.

To simplify the given rational expression means we have to convert the given expression into an expression that is easy to understand.

Given

{eq}\dfrac{5y^2}{5y^2+30y-35}.\dfrac{2y^3+30y^2+112y}{8y+44} {/eq}

We have to multiply the rational expressions and simplify the result.

{eq}\begin{align} \dfrac{5y^2}{5y^2+30y-35}\cdot\dfrac{2y^3+30y^2+112y}{8y+44} &=\dfrac{5y^2}{5(y^2+6y-7)}\cdot \dfrac{2y(y^2+15y+56}{4(2y+11)}\\ &=\dfrac{y^2}{(y^2+6y-7)}\cdot \dfrac{y(y^2+15y+56)}{2(2y+11)}\\ &=\dfrac{y^2}{(y^2+7y-y-7)}\cdot \dfrac{y(y^2+8y+7y+56)}{2(2y+11)}\ & \left [ \text{Splitting the middle term} \right ]\\ &=\dfrac{y^2}{y(y+7)-1(y+7)}\cdot \dfrac{y\left ( y(y+8)+7(y+8) \right )}{2(2y+11)}\\ &=\dfrac{y^2}{(y-1)(y+7)}\cdot \dfrac{y(y+7)(y+8)}{2(2y+11)}\\ &=\dfrac{y^3(y+8)}{2(y-1)(2y+11)}\\ \end{align} {/eq}

Now, we have to find the values of {eq}y {/eq} that makes the resulting expression undefined.

The resulting expression becomes undefined when the denominator of the expression becomes zero i.e

{eq}\begin{align} (y-1)(2y+11) &=0\\ y-1=0 \ & or \ \quad\: \ 2y+11=0\\ y=1 \ & \ or \ \quad\: \ y=\dfrac{-11}{2} \end{align} {/eq}

So, the resulting expression becomes undefined when {eq}\color{blue}{\boxed{y=1 \ , \dfrac{-11}{2}}} {/eq}

The correct answer is option {eq}\color{blue}{C, D} {/eq} 