How do you find the quotient and remainder when x^3 -5x^2+px+6 is divided by x+2?

Mar 4, 2017

Quotient is ${x}^{2} - 7 x + p + 14$ and remainder is $- 2 p - 22$

Explanation:

Let us divide ${x}^{3} - 5 {x}^{2} + p x + 6$ by $x + 2$ by synthetic division

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

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

Two As $x + 2 = 0$ gives $x = - 2$ put $- 2$ at the left.

$- 2 | \textcolor{w h i t e}{X} 1 \text{ "color(white)(X)-5color(white)(XX)p" "" } 6$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X X}$
" "stackrel("—————————————)

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

$- 2 | \textcolor{w h i t e}{X} 1 \text{ "color(white)(X)-5color(white)(XX)p" "" } 6$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)
$\textcolor{w h i t e}{\times} | \textcolor{w h i t e}{X} \textcolor{red}{1}$

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

$- 2 | \textcolor{w h i t e}{X} 1 \text{ "color(white)(X)-5color(white)(XX)p" "" } 6$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{\times X} - 2$
" "stackrel("—————————————)
$\textcolor{w h i t e}{\times} | \textcolor{w h i t e}{X} \textcolor{b l u e}{1}$

$- 2 | \textcolor{w h i t e}{X} 1 \text{ "color(white)(X)-5color(white)(XX)p" "" } 6$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{x X X} - 2$
" "stackrel("—————————————)
$\textcolor{w h i t e}{\times} | \textcolor{w h i t e}{X} \textcolor{b l u e}{1} \textcolor{w h i t e}{X 11} \textcolor{red}{-} 7$

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

$- 2 | \textcolor{w h i t e}{X} 1 \text{ "color(white)(X)-5color(white)(XX)p" "" } \textcolor{w h i t e}{X X} 6$
$\textcolor{w h i t e}{\times} | \text{ } \textcolor{w h i t e}{X X x} - 2 \textcolor{w h i t e}{\times} 14 \textcolor{w h i t e}{X} - 2 p - 28$
" "stackrel("—————------------————————)
$\textcolor{w h i t e}{\times} | \textcolor{w h i t e}{X} \textcolor{b l u e}{1} \textcolor{w h i t e}{X X} \textcolor{red}{-} 7 \textcolor{w h i t e}{X X} \textcolor{red}{p + 14} \textcolor{w h i t e}{} \textcolor{red}{\left(- 2 p - 22\right)}$

Hence, Quotient is ${x}^{2} - 7 x + p + 14$ and remainder is $- 2 p - 22$.

We can also work out remainder using remainder theorem, which gives remainder as $f \left(- 2\right)$ i.e.

$f \left(- 2\right) = {\left(- 2\right)}^{3} - 5 {\left(- 2\right)}^{2} + p \left(- 2\right) + 6 = - 8 - 20 - 2 p + 6 = - 2 p - 22$