How do you use synthetic division to divide 4x^3 + 7x^2 – 13x – 3 by x-7?

Jul 31, 2015

$\frac{4 {x}^{3} + 7 {x}^{2} - 13 x - 3}{x - 7} = \textcolor{b l u e}{4 {x}^{2} + 35 x + 232} + \frac{\textcolor{red}{1621}}{x - 7}$

Explanation:

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

$| 4 \text{ "" "7color(white)(1)-13" "" } - 3$
$| \textcolor{w h i t e}{1}$
stackrel("—————————————————)

Step 2. Put the divisor at the left.

$\textcolor{red}{7} | 4 \text{ "" "7color(white)(1)-13" "" } - 3$
$\textcolor{w h i t e}{1} | \textcolor{w h i t e}{1}$
" "stackrel("—————————————————)

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

$7 | 4 \text{ "" "7color(white)(1)-13" "" } - 3$
$\textcolor{w h i t e}{1} | \textcolor{w h i t e}{1}$
" "stackrel("—————————————————)
$\textcolor{w h i t e}{1} | \textcolor{red}{4}$

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

$7 | 4 \text{ "" "7color(white)(1)-13" "" } - 3$
$\textcolor{w h i t e}{1} | \text{ "" } \textcolor{red}{28}$
" "stackrel("—————————————————)
$\textcolor{w h i t e}{1} | 4$

Step 5. Add down the column.

$7 | 4 \text{ "" "7color(white)(1)-13" "" } - 3$
$\textcolor{w h i t e}{1} | \text{ "" } 28$
" "stackrel("—————————————————)
$\textcolor{w h i t e}{1} | 4 \text{ } \textcolor{w h i t e}{1} \textcolor{red}{35}$

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

$7 | 4 \text{ "" "7color(white)(1)-13" "" } - 3$
$\textcolor{w h i t e}{1} | \text{ "" "28" "color(white)(1)245" } \textcolor{w h i t e}{1} 1624$
" "stackrel("—————————————————)
color(blue)(color(white)(1)|4" "color(white)(1)35" "color(white)(1)232)" "color(white)(1)color(red)(1621)

$\frac{4 {x}^{3} + 7 {x}^{2} - 13 x - 3}{x - 7} = 4 {x}^{2} + 35 x + 232 + \frac{1621}{x - 7}$

Check:

$\left(x - 7\right) \left(4 {x}^{2} + 35 x + 232 + \frac{1621}{x - 7}\right) = \left(x - 7\right) \left(4 {x}^{2} + 35 x + 232\right) - 1621 = 4 {x}^{3} + 35 {x}^{2} + 232 x - 28 {x}^{2} - 245 x - 1624 + 1621$

$= 4 {x}^{3} + 7 {x}^{2} - 13 x - 3$