Chapter 7.9, Problem 7MRE

To determine

To calculate: The simplified form of the expression $\frac{4x^2+2x+36}{x+4}$ .

Answer to Problem 7MRE

The simplified form of the expression $\frac{4x^2+2x+36}{x+4}$ is $\left(4x-14\right)+\frac{92}{\left(x+4\right)}$ .

Explanation of Solution

Given information:

The expression $\frac{4x^2+2x+36}{x+4}$ .

Formula used:

When a polynomial is divided by its factor then dividend is the product of divisor and quotient increased by remainder.

Calculation:

Consider the expression $\frac{4x^2+2x+36}{x+4}$ .

When a polynomial is divided by its factor then dividend is the product of divisor and quotient increased by remainder.

Apply the method of long division. First multiply the expression $x+4$ by $4x$ . Subtract it from the dividend. Next multiply $x+4$ by $-14$ . Subtract the expression obtained.

  $\begin{array}{l}x+4\begin{array}{c}\hfill 4x-14\\ \hfill \overline{)4x^2+2x+26}\end{array}\\ \begin{array}{ccc}& & 4x^2+16x\end{array}\\ \begin{array}{ccc}& & \begin{array}{cc}& -14x+36\end{array}\end{array}\\ \begin{array}{cc}& \begin{array}{ccc}& & -14x-56\end{array}\end{array}\\ \begin{array}{cccccc}& & & & & 92\end{array}\end{array}$

Now, since $\text{Dividend}=\text{Quotient}\times \text{Divisor}+\text{Remainder}$ . Here, dividend is $4x^2+2x+36$ quotient is $4x-14$ , divisor is $x+4$ and remainder is $92$ .

Therefore, $4x^2+2x+36=\left(x+4\right)\left(4x-14\right)+92$ that is,

  $\frac{4x^2+2x+36}{x+4}=\left(4x-14\right)+\frac{92}{\left(x+4\right)}$

Thus, the simplified form of the expression $\frac{4x^2+2x+36}{x+4}$ is $\left(4x-14\right)+\frac{92}{\left(x+4\right)}$ .