Chapter 4.3, Problem 10E

To determine

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

Answer to Problem 10E

The result is $4x^2+12x+44+\frac{156x+84}{x^2-3x-2}$ .

Explanation of Solution

Given information:

Given expression,

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

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 $4x^2$ and subtract.

Multiply $4x^2$ by $x^2-3x-2$ which is $4x^2\left(x^2-3x-2\right)=4x^4-12x^3-8x^2$ and subtract.

Subtract and bring down $5x-4$ .

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

Multiply $12x$ by $x^2-3x-2$ which is $12x\left(x^2-3x-2\right)=12x^3-36x^2-24x$ and subtract.

Divide $44x^2$ by $x^2$ which is 44 and subtract.

Multiply 44 by $x^2-3x-2$ which is $44\left(x^2-3x-2\right)=44x^2-132x-88$ and subtract.

  $x^2-3x-2\begin{array}{c}\hfill 4x^2+12x+44\\ \hfill \overline{)\begin{array}{l}4x^4+0\cdot x^3+0\cdot x^2+5x-4\\ \underset{\_}{-\left(4x^4-12x^3-8x^2\right)}\\ 12x^3+8x^2+5x-4\\ \underset{\_}{-\left(12x^3-36x^2-24x\right)}\\ 44x^2+24x-4\\ \underset{\_}{-\left(44x^2-132x-88\right)}\\ 156x+84\end{array}}\end{array}$

Thus, the result is $4x^2+12x+44+\frac{156x+84}{x^2-3x-2}$ .