# How do you divide (8r^3-55r^2+44r-12)div(r-6) using synthetic division?

Aug 21, 2016

$\left(8 {r}^{3} - 55 {r}^{2} + 44 r - 12\right) \div \left(r - 6\right) = 8 {r}^{2} - 7 r + 2$

#### Explanation:

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

(8r^3-55r^2+44r-12)div(r-6) = ?????????
" (dividend) " div " (divisor)" = ("quotient")

$\textcolor{m a \ge n t a}{\text{step 1:}}$ The dividend must be in descending powers of r.
color(white)(xxxxxxxxxx)8r^3 " -55r^2 +44r -12
$\textcolor{w h i t e}{\times \times \times \times} \Rightarrow 8 \text{ -55 +44 -12}$

Only use the numerical coefficients in the top row.

(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. $\text{ " (r-6) = 0 rArr r = color(orange)(6) " this goes outside}$

color(white)(xxxxx) | color(brown)(8)" "-55" "+44 " "-12 color(magenta)(" step 1")
$\textcolor{w h i t e}{\times} \textcolor{\mathmr{and} a n \ge}{6} \text{ "| darr " "color(red)(48) " "color(blue)(-42) " } \textcolor{o l i v e}{12}$
$\textcolor{w h i t e}{\times \times \times} \underline{\text{ }}$
$\textcolor{w h i t e}{\times \times \times x} \textcolor{b r o w n}{8} \text{ "color(blue)(-7) " "color(olive)(+2)" "color(teal)(0) larr " no remainder!}$

$\textcolor{w h i t e}{\times \times . \times} \uparrow \text{ "uarr " } \uparrow$
$\textcolor{w h i t e}{\times \times \times x} {r}^{2} \text{ "r^1 " } {r}^{0}$

Step 3 : Begin the division:

$\text{Bring down the " color(brown)( 8 ) " to below the line}$
$\text{multiply } \textcolor{\mathmr{and} a n \ge}{6} \times \textcolor{b r o w n}{8} = \textcolor{red}{48}$
$\text{Add } - 55 + \textcolor{red}{48} = \textcolor{b l u e}{- 7}$
$\text{multiply } \textcolor{\mathmr{and} a n \ge}{6} \times \textcolor{b l u e}{- 7} = \textcolor{b l u e}{- 42}$
$\text{Add } 44 \textcolor{b l u e}{- 42} = \textcolor{o l i v e}{+ 2}$
$\text{multiply } \textcolor{\mathmr{and} a n \ge}{6} \times \textcolor{o l i v e}{2} = \textcolor{o l i v e}{12}$
$\text{Add } - 12 + \textcolor{o l i v e}{12} = \textcolor{t e a l}{0}$

That's it Folks!

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

We divided an expression with ${r}^{3}$ by an expression with $r$,
so the first term will be ${r}^{3} / r = {r}^{2}$

The last value is the remainder. In this case it is $\textcolor{t e a l}{0}$

This means that $\left(r - 6\right)$ is a factor of $8 {r}^{3} - 55 {r}^{2} + 44 r - 12$

$\left(8 {r}^{3} - 55 {r}^{2} + 44 r - 12\right) \div \left(r - 6\right) = 8 {r}^{2} - 7 r + 2 \text{ rem 0}$