# How do you divide (r^2+6r+15)div(r+5) using synthetic division?

Nov 6, 2016

Remainder is $10$ and $\left({r}^{2} + 6 r + 15\right) \div \left(r + 5\right) = r + 1 + \frac{10}{r + 5}$

#### Explanation:

To divide ${r}^{2} + 6 r + 15$ by $r + 5$

One Write the coefficients of $t$ in the dividend inside an upside-down division symbol.

$\textcolor{w h i t e}{1} | \textcolor{w h i t e}{X} 1 \text{ } \textcolor{w h i t e}{X} 6 \textcolor{w h i t e}{X X} 15$
$\textcolor{w h i t e}{1} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)

Two Put the divisor at the left (as it is $r + 5$, we need to put $- 5$.

$- 5 | \textcolor{w h i t e}{X} 1 \text{ } \textcolor{w h i t e}{X} 6 \textcolor{w h i t e}{X X} 15$
$\textcolor{w h i t e}{X x} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)

Three Drop the first coefficient of the dividend below the division symbol.

$- 5 | \textcolor{w h i t e}{X} 1 \text{ } \textcolor{w h i t e}{X} 6 \textcolor{w h i t e}{X X} 15$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)
$\textcolor{w h i t e}{} \left(1\right) | \textcolor{w h i t e}{X} \textcolor{red}{1}$

Four Multiply the result by the constant, and put the product in the next column.

$- 5 | \textcolor{w h i t e}{X} 1 \text{ } \textcolor{w h i t e}{X} 6 \textcolor{w h i t e}{X X} 15$
$\textcolor{w h i t e}{221} | \text{ } \textcolor{w h i t e}{X} - 5$
" "stackrel("—————————————)
$\textcolor{w h i t e}{221} | \textcolor{w h i t e}{X} \textcolor{b l u e}{1}$

$- 5 | \textcolor{w h i t e}{X} 1 \text{ } \textcolor{w h i t e}{X} 6 \textcolor{w h i t e}{X X} 15$
$\textcolor{w h i t e}{221} | \text{ } \textcolor{w h i t e}{X 1} - 5$
" "stackrel("—————————————)
$\textcolor{w h i t e}{221} | \textcolor{w h i t e}{X} \textcolor{b l u e}{1} \textcolor{w h i t e}{X X} \textcolor{red}{1}$

Six Repeat Steps Four and Five until you can go no farther.

$2 | \textcolor{w h i t e}{X} 1 \text{ } \textcolor{w h i t e}{X} 6 \textcolor{w h i t e}{X X} 15$
$\textcolor{w h i t e}{1} | \text{ } \textcolor{w h i t e}{X 1} - 5 \textcolor{w h i t e}{X} - 5$
" "stackrel("—————————————)
$\textcolor{w h i t e}{1} | \textcolor{w h i t e}{X} \textcolor{b l u e}{1} \textcolor{w h i t e}{X 11} \textcolor{red}{1} \textcolor{w h i t e}{X X} \textcolor{red}{10}$

Remainder is $10$ and

$\left({r}^{2} + 6 r + 15\right) \div \left(r + 5\right) = r + 1 + \frac{10}{r + 5}$