# How do you divide (x^3-11x^2+22x+40)div(x-5) using synthetic division?

Sep 18, 2016

${x}^{2} - 6 x - 8$

#### Explanation:

The coefficients of the first polynomial become the dividend.
The "zero" of the second polynomial becomes the divisor. In other words, solve $x - 5 = 0$ and use the result (5) as the divisor.

$5 \rfloor 1 \textcolor{w h i t e}{a a} - 11 \textcolor{w h i t e}{a a a a} 22 \textcolor{w h i t e}{a a a} 40$
$\textcolor{w h i t e}{a a A a a a a a} 5 \textcolor{w h i t e}{a} - 30 \textcolor{w h i t e}{a} - 40$
$\textcolor{w h i t e}{a a}$-----------------------------------
$\textcolor{w h i t e}{a a} \textcolor{red}{1} \textcolor{w h i t e}{a a a} \textcolor{red}{- 6} \textcolor{w h i t e}{a a} \textcolor{red}{- 8} \textcolor{w h i t e}{a a a a} \textcolor{red}{0}$

To do synthetic division, "pull down" the first coefficient ($1$) below the line. Multiply the divisor by the first coefficient. $5 \cdot 1 = 5$

Write the product ($5$) under the next coefficient and add. ($- 11 + 5 = - 6$) Write the sum under the line.

Multiply the divisor by the new number under the line.
($5 \cdot - 6 = - 30$)

Write the product (-30) under the next coefficient and add $22 + - 30 = - 8$

Continue as above. The remainder is zero.

Then, use the numbers you've written under the line as the coefficients of the quotient. Start with a variable of degree one less than the degree of the dividend.

The quotient is:
$\textcolor{red}{1} {x}^{2} \textcolor{red}{- 6} x \textcolor{red}{- 8} + \frac{\textcolor{red}{0}}{{x}^{3} - 11 {x}^{2} + 22 x + 40}$

The last term represents the remainder and is equal to zero.

Thus, the quotient is:
${x}^{2} - 6 x - 8$