# How do you divide (x^4+5x^2+2)/(x+2)?

Feb 16, 2017

$\frac{{x}^{4} + 5 {x}^{2} + 2}{x + 2} = {x}^{3} - 2 {x}^{2} + 9 x - 18 \text{ }$rem 38

#### Explanation:

The process is fairly simple and is a repetition of the same sequence again and again. Describing the process is not easy.....

Terminology first:
The dividend is divided by the divisor to get the quotient.

$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots . .} \underline{\text{quotient}}$
$\text{divisor "|"dividend}$

Algebraic long division follows the same pattern as long division in arithmeic: here are the steps...

(1.) Write the dividend in decreasing powers of the variable ($w$).
Make sure there is a space for any missing powers.

(2.) Divide the first term of the divisor into the term of the dividend with the highest power available and write the answer at the top as the quotient.

(3.) Multiply that answer by both terms in the divisor and write the answers under the like terms in the dividend.

(4.) Subtract (change signs) and bring down the next highest power.

REPEAT steps 2 to 5.

$\textcolor{w h i t e}{\ldots \ldots \ldots .} \underline{\textcolor{w h i t e}{\ldots \ldots \ldots \ldots}} \underline{{x}^{3} - 2 {x}^{2} + 9 x - 18}$
$x + 2 | {x}^{4} + 0 {x}^{3} + 5 {x}^{2} + 0 x + 2$
color(white)(............)ul(x^4+2x^3
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots} - 2 {x}^{3} + 5 {x}^{2}$
color(white)(................)ul(-2x^3-4x^2
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} 9 {x}^{2}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} \underline{9 {x}^{2} + 18 x}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} - 18 x + 2$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots} \underline{- 18 x - 36}$
$\textcolor{w h i t e}{\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots .} 38$

$\frac{{x}^{4} + 5 {x}^{2} + 2}{x + 2} = {x}^{3} - 2 {x}^{2} + 9 x - 18 \text{ }$rem 38