# How do you divide (x^2-3x-18)div(x-6) using synthetic division?

Aug 21, 2016

$\left({x}^{2} - 3 x - 18\right) \div \left(x - 6\right) = x + 3 , \text{rem } 0$

#### Explanation:

The method is easy, but the format is difficult to show. I'll do my best.
$\left({x}^{2} - 3 x - 18\right) \div \left(x - 6\right)$
$\text{ (dividend) " div " (divisor)}$

$\textcolor{m a \ge n t a}{\text{step 1:}}$ Dividend must be in descending powers of x.
$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times \times \times \times} {x}^{2} \text{ "-3x" } - 18$
Only use the numerical coefficients $\Rightarrow 1 \text{ "-3" "-18 }$

(If there are any missing, leave a space or fill in a zero).

$\textcolor{\mathmr{and} a n \ge}{\text{Step 2}}$: Make the divisor = 0. $\text{ "(x-6) = 0 rArr x = color(orange)(6) " this goes outside}$

color(white)(xxxxxxxxx) | color(brown)(1)" "-3" "-18 " "color(magenta)("step 1")
$\textcolor{w h i t e}{\times \times \times} \textcolor{\mathmr{and} a n \ge}{6} \text{ "| darr " "color(red)(6) " } \textcolor{b l u e}{18}$
$\textcolor{w h i t e}{\times \times \times \times \times} \underline{\text{ }}$
$\textcolor{w h i t e}{\times \times \times \times \times x} \textcolor{b r o w n}{1} \text{ "color(blue)(3) " "color(teal)(0) larr "no remainder!}$

$\textcolor{w h i t e}{\times \times . . \times \times x} \uparrow \text{ } \uparrow$
$\textcolor{w h i t e}{\times \times \times \times \times x} x \text{ } {x}^{0}$

Step 3: Begin the division:
-"Bring down the " color(brown)( 1 ) " to below the line"
-$\text{multiply } \textcolor{\mathmr{and} a n \ge}{6} \times \textcolor{b r o w n}{1} = \textcolor{red}{6}$
$\text{Add} - 3 + \textcolor{red}{6} = \textcolor{b l u e}{3}$
-$\text{multiply } \textcolor{\mathmr{and} a n \ge}{6} \times \textcolor{b l u e}{3} = \textcolor{b l u e}{18}$
$\text{Add} - 18 + \textcolor{b l u e}{18} = \textcolor{t e a l}{0}$

That's it Folks!

We have now found the numerical coefficients of the terms in the quotient (answer)

We divided an expression with ${x}^{2}$ by an expression with $x$,
so the first term will be ${x}^{2} / x = x$

The last value is the remainder. In this case it is $\textcolor{t e a l}{0}$
This means that $x - 6$ is a factor of ${x}^{2} - 3 x - 18$

$\left({x}^{2} - 3 x - 18\right) \div \left(x - 6\right) = \textcolor{b r o w n}{1} x + \textcolor{b l u e}{3} , \text{rem } \textcolor{t e a l}{0}$