# How do you use synthetic substitution to factor x^3+9x^2+24x+20 if x+5 is one of its factor?

Oct 3, 2015

${x}^{3} + 9 {x}^{2} + 24 x + 20 = \textcolor{b l u e}{\left(x + 5\right) \left(x + 2\right) \left(x + 2\right)}$

#### Explanation:

If $x + 5$ is a factor, your divisor in synthetic substitution is $- 5$.

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

$\textcolor{w h i t e}{X l l} | 1 \textcolor{w h i t e}{X l l} 9 \textcolor{w h i t e}{X X} 24 \textcolor{w h i t e}{X X l} 20$
$\textcolor{w h i t e}{X X} |$
color(white)(XX)stackrel("—————————————)

Step 2. Put the divisor at the left.

$\textcolor{b l u e}{- 5} | 1 \textcolor{w h i t e}{X l l} 9 \textcolor{w h i t e}{X X} 24 \textcolor{w h i t e}{X X l} 20$
$\textcolor{w h i t e}{X X} |$
color(white)(XX)stackrel("—————————————)

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

$- 5 | 1 \textcolor{w h i t e}{X l l} 9 \textcolor{w h i t e}{X X} 24 \textcolor{w h i t e}{X X l} 20$
$\textcolor{w h i t e}{X X} |$
color(white)(XX)stackrel("—————————————)
$\textcolor{w h i t e}{X l l} | \textcolor{b l u e}{1}$

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

$- 5 | 1 \textcolor{w h i t e}{X l l} 9 \textcolor{w h i t e}{X X} 24 \textcolor{w h i t e}{X X l} 20$
$\textcolor{w h i t e}{X l l} | \textcolor{w h i t e}{X l} \textcolor{b l u e}{- 5}$
color(white)(XX)stackrel("—————————————)
$\textcolor{w h i t e}{X l l} | 1$

Step 5. Add down the column.

$- 5 | 1 \textcolor{w h i t e}{X l l} 9 \textcolor{w h i t e}{X X} 24 \textcolor{w h i t e}{X X l} 20$
$\textcolor{w h i t e}{X l l} | \textcolor{w h i t e}{l l} - 5$
color(white)(XX)stackrel("—————————————)
$\textcolor{w h i t e}{X l l} | 1 \text{ } \textcolor{w h i t e}{l} \textcolor{b l u e}{4}$

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

$- 5 | 1 \textcolor{w h i t e}{X l l} 9 \textcolor{w h i t e}{X X} 24 \textcolor{w h i t e}{X X l} 20$
$\textcolor{w h i t e}{X l l} | \textcolor{w h i t e}{l l} - 5 \textcolor{w h i t e}{1} - 20 \textcolor{w h i t e}{1} - 20$
color(white)(XX)stackrel("————————————)
$\textcolor{w h i t e}{X l l} | 1 \text{ "color(white)(l)4" "color(white)(Xl)4" } \textcolor{w h i t e}{X l} \textcolor{red}{0}$

$\frac{{x}^{3} + 9 {x}^{2} + 24 x + 20}{x + 5} = {x}^{2} + 4 x + 4$

You can factor the quadratic as ${x}^{2} + 4 x + 4 = \left(x + 2\right) \left(x + 2\right)$

So

${x}^{3} + 9 {x}^{2} + 24 x + 20 = \left(x + 5\right) \left(x + 2\right) \left(x + 2\right)$