Chapter 4.4, Problem 10Q

To determine

To find: the divide the expression $4x^4-2x^3+x^2-5x+8$ by $x^2-2x-1$ using long division of polynomial.

Answer to Problem 10Q

The result is $4x^2+6x+17+\frac{35x+25}{x^2-2x-1}$ .

Explanation of Solution

Given information:

Given expression,

  $\left(4x^4-2x^3+x^2-5x+8\right)÷\left(x^2-2x-1\right)$

Calculation:

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

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

Multiply $4x^2$ by $x^2-2x-1$ which is $4x^2\left(x^2-2x-1\right)=4x^4-8x^3-4x^2$ and subtract.

Subtract and bring down $-5x+8$ .

Divide $6x^3$ by $x^2$ which is $6x$ .

Multiply $6x$ by $x^2-2x-1$ which is $6x\left(x^2-2x-1\right)=6x^3-12x^2-6x$ and subtract.

Divide $17x^2$ by $x^2$ which is 17.

Multiply 17 by $x^2-2x-1$ which is

  $17\left(x^2-2x-1\right)=17x^2-34x-17$ and subtract.

  $x^2-2x-1\begin{array}{c}\hfill 4x^2+6x+17\\ \hfill \overline{)\begin{array}{l}4x^4-2x^3+x^2-5x+8\\ \underset{\_}{-\left(4x^4-8x^3-4x^2\right)}\\ 6x^3+5x^2-5x+8\\ \underset{\_}{-\left(6x^3-12x^2-6x\right)}\\ 17x^2+x+8\\ \underset{\_}{-\left(17x^2-34x-17\right)}\\ 35x+25\end{array}}\end{array}$

Thus, the result is $4x^2+6x+17+\frac{35x+25}{x^2-2x-1}$ .