# How do you divide (x^2+2x+15)div(x-3) using synthetic division?

Aug 21, 2016

$\left({x}^{2} + 2 x + 15\right) \div \left(x - 3\right) = \left(x + 5\right) \text{ rem 30}$

#### Explanation:

The method is easy, but the process is a bit difficult to explain.

$\text{ } \left({x}^{2} + 2 x + 15\right) \div \left(x - 3\right)$
$\text{ (dividend) " div " (divisor)}$

$\textcolor{m a \ge n t a}{\text{step 1:}}$ The dividend must be in descending powers of x.
$\textcolor{w h i t e}{\times \times \times \times \times \times \times \times \times \times \times \times} {x}^{2} \text{ "+2x" } + 15$
Only use the numerical coefficients $\Rightarrow 1 \text{ "+2" "+15 }$

(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. $\text{ "(x-3) = 0 rArr x = color(orange)(3) " this goes outside}$

color(white)(xxxxxxxxx) | color(brown)(1)" "+2" "+15" "color(magenta)("step 1")
$\textcolor{w h i t e}{\times \times \times} \textcolor{\mathmr{and} a n \ge}{3} \text{ "| darr " "color(red)(3) " } \textcolor{b l u e}{15}$
$\textcolor{w h i t e}{\times \times \times \times \times} \underline{\text{ }}$
$\textcolor{w h i t e}{\times \times \times \times \times x} \textcolor{b r o w n}{1} \text{ "color(blue)(5) " "color(teal)(30) larr "remainder!}$

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

Step 3: Begin the division:

$\text{Bring down the " color(brown)( 1 ) " to below the line}$
$\text{multiply } \textcolor{\mathmr{and} a n \ge}{3} \times \textcolor{b r o w n}{1} = \textcolor{red}{3}$
$\text{Add } 2 + \textcolor{red}{3} = \textcolor{b l u e}{5}$
$\text{multiply } \textcolor{\mathmr{and} a n \ge}{3} \times \textcolor{b l u e}{5} = \textcolor{b l u e}{15}$
$\text{Add} 15 + \textcolor{b l u e}{15} = \textcolor{t e a l}{30}$

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 $\textcolor{t e a l}{30}$
This means that $x - 3$ is not a factor of ${x}^{2} + 2 + 15$

$\left({x}^{2} + 2 x + 15\right) \div \left(x - 3\right) = \textcolor{b r o w n}{1} x + \textcolor{b l u e}{5} , \text{rem } \textcolor{t e a l}{30}$