# How do you use the remainder theorem and synthetic division to find the remainder when x^3 - 2x^2 + 5x - 6 div x - 3?

Sep 22, 2015

Here is the synthetic division.

#### Explanation:

(Division format from Ernest Z. here on Socratic)

Step 1. Write only the coefficients of $x$ in the dividend inside an upside-down division symbol.

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

Step 2. Put the constant in the divisor $x - c$ at the left. In this case $c = 3$

$3 | 1 \text{ "-2color(white)(XX)5" "" } - 6$
$\textcolor{w h i t e}{1} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)

Step 3. Drop the first coefficient of the dividend below the division symbol.

$3 | 1 \text{ "-2color(white)(XX)5" "" } - 6$
$\textcolor{w h i t e}{1} | \text{ } \textcolor{w h i t e}{X}$
" "stackrel("—————————————)
$\textcolor{w h i t e}{1} | 1$

Step 4. Multiply the result by the constant, and put the product in the next column.

$3 | 1 \text{ "-2color(white)(XX)5" "" } - 6$
$\textcolor{w h i t e}{1} | \text{ } \textcolor{w h i t e}{X 1} 3$
" "stackrel("—————————————)
$\textcolor{w h i t e}{1} | 1$

Step 5. Add down the column.

$3 | 1 \text{ "-2color(white)(XX)5" "" } - 6$
$\textcolor{w h i t e}{1} | \text{ "color(white)(X1)3" }$
" "stackrel("—————————————)
$\textcolor{w h i t e}{1} | 1 \text{ "" } 1$

Step 6. Repeat Steps 4 and 5 until you can go no farther.

$3 | 1 \text{ "-2color(white)(XX)5" } \textcolor{w h i t e}{1} - 6$
$\textcolor{w h i t e}{1} | \text{ "color(white)(X1)3" } \textcolor{w h i t e}{X 1} 3 \textcolor{w h i t e}{X X 1} 24$
" "stackrel("—————————————)
$\textcolor{w h i t e}{1} | 1 \text{ "color(white)(X)1" "color(white)(X1)8" ""|} \textcolor{w h i t e}{X 1} 18$

The remainder is $18$

(And the quotient is $1 {x}^{2} + 1 x + 8$, more commonly written ${x}^{2} + x + 8$)