How do you use synthetic division to divide (5x^3+18x^2+7x-6)div(x+3)?

Oct 5, 2017

$5 {x}^{2} + 3 x - 2$

Explanation:

Oct 5, 2017

Quotient is $5 {x}^{2} + 3 x - 2$ andremainder is $0$. See the process below.

Explanation:

To divide $f \left(2\right)$ for $5 {x}^{3} + 18 {x}^{2} + 7 x - 6$ by $x + 3$

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

$\textcolor{w h i t e}{1} | \textcolor{w h i t e}{X} 5 \text{ "color(white)(X)18color(white)(XX)+7" "" } - 6$
$\textcolor{w h i t e}{1} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)

Two Put $- 3$ in the divisor at the left as $x + 3 = 0$ gives $x = - 3$

$- 3 | \textcolor{w h i t e}{X} 5 \text{ "color(white)(X)18color(white)(XX)7" "" } - 6$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)

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

$- 3 | \textcolor{w h i t e}{X} 5 \text{ "color(white)(X)18color(white)(XX)7" "" } - 6$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)
" "color(white)(1)|color(white)(X)color(red)

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

$- 3 | \textcolor{w h i t e}{X} 5 \text{ "color(white)(X)18color(white)(XX)7" "" } - 6$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)
$\text{ } \textcolor{w h i t e}{1} | \textcolor{w h i t e}{X} \textcolor{b l u e}{5}$

$- 3 | \textcolor{w h i t e}{X} 5 \text{ "color(white)(X)18color(white)(XX)7" "" } - 6$
$\text{ "color(white)(1)|" } \textcolor{w h i t e}{X} - 15$
color(white)(1)stackrel("—————————————)
$\textcolor{w h i t e}{\times} | \textcolor{w h i t e}{X} \textcolor{b l u e}{5} \textcolor{w h i t e}{X X X} \textcolor{red}{3}$

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

$- 3 | \textcolor{w h i t e}{X} 5 \text{ "color(white)(X)18color(white)(XX)7" "" } - 6$
$\text{ "color(white)(1)|" } \textcolor{w h i t e}{X} - 15 \textcolor{w h i t e}{X} - 9 \textcolor{w h i t e}{X X X} 6$
color(white)(1)stackrel("—————————————)
$\textcolor{w h i t e}{\times} | \textcolor{w h i t e}{X} \textcolor{b l u e}{5} \textcolor{w h i t e}{X X X} \textcolor{red}{3} \textcolor{w h i t e}{X} \textcolor{red}{-} 2 \textcolor{w h i t e}{X X X} \textcolor{red}{0}$

Hence remainder is $0$ and quotient is $5 {x}^{2} + 3 x - 2$