Chapter 4.3, Problem 9E

To determine

To find: the given division using long division of polynomial.

Answer to Problem 9E

The result is $5x^2-12x+37+\frac{-122x+109}{x^2+2x-4}$ .

Explanation of Solution

Given information:

Given expression,

  $\left(5x^4-2x^3-7x^2-39\right)÷\left(x^2+2x-4\right)$

Calculation:

To write the numerator in standard form writing any missing term with a coefficient of zero.

Divide $5x^4$ by $x^2$ which is $5x^2$ .

Multiply $5x^2$ by $x^2+2x-4$ which is $5x^2\left(x^2+2x-4\right)=5x^4+10x^3-20x^2$ .

Subtract and bring down $-39$ .

Divide $-12x^3$ by $x^2$ which is $-12x$ and subtract.

Multiply $-12x$ by $x^2+2x-4$ which is $-12x\left(x^2+2x-4\right)=-12x^3-24x^2+48x$ and subtract.

Divide $37x^2$ by $x^2$ which is 37 and subtract.

Multiply 37 by $x^2+2x-4$ which is $37\left(x^2+2x-4\right)=37x^2+74x-148$ and subtract.

  $x^2+2x-4\begin{array}{c}\hfill 5x^2-12x+37\\ \hfill \overline{)\begin{array}{l}5x^4-2x^3-7x^2-39\\ \underset{\_}{-\left(5x^4+10x^3-20x^2\right)}\\ -12x^3+13x^2-39\\ \underset{\_}{-\left(-12x^3-24x^2+48x\right)}\\ 37x^2-48x-39\\ \underset{\_}{-\left(37x^2+74x-148\right)}\\ -122x+109\end{array}}\end{array}$

Thus, the result is $5x^2-12x+37+\frac{-122x+109}{x^2+2x-4}$ .