Chapter 4.3, Problem 8E

To determine

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

Answer to Problem 8E

The result is $7x+1+\frac{-6x-1}{x^2+1}$ .

Explanation of Solution

Given information:

Given expression,

  $\left(7x^3+x^2+x\right)÷\left(x^2+1\right)$

Calculation:

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

Divide $7x^3$ by $x^2$ which is $7x$ . Multiply $7x$ by $x^2+1$ which is $7x\left(x^2+1\right)=7x^3+7x$ .

Subtract and bring down $x$ .

Divide $x^2$ by $x^2$ which is 1. Multiply 1 by $x^2+1$ which is $1\left(x^2+1\right)=x^2+1$ and subtract.

  $x^2+1\begin{array}{c}\hfill 7x+1\\ \hfill \overline{)\begin{array}{l}7x^3+x^2+x\\ \underset{\_}{-\left(7x^3+x\right)}\\ x^2-6x\\ \underset{\_}{-\left(x^2+1\right)}\\ -6x-1\end{array}}\end{array}$

Thus, the result is $7x+1+\frac{-6x-1}{x^2+1}$ .