# How do you use synthetic division to find the factors of x^3 - 3x^2 - 10x + 24?

Jun 8, 2015

First use the rational roots theorem to provide a list of possible roots of ${x}^{3} - 3 {x}^{2} - 10 x + 24 = 0$. Having found one, use synthetic division by the corresponding factor to simplify the problem.

#### Explanation:

Write $f \left(x\right) = {x}^{3} - 3 {x}^{2} - 10 x + 24$

The rational roots theorem tells us that any rational root $\frac{p}{q}$ of $f \left(x\right) = 0$, expressed in simplest terms will have the property that $p$ is a divisor of the constant term ($24$) and $q$ is a divisor of the coefficient ($1$) of the highest power of $x$.

So the only possible rational roots of $f \left(x\right) = 0$ are the integer factors of $24$:

$\pm 1$, $\pm 2$, $\pm 3$, $\pm 4$, $\pm 6$, $\pm 8$, $\pm 12$, $\pm 24$

Aside: If all the roots are rational then they are 3 integers whose product is $- 24$, so at least one of them must have absolute value less than or equal to $\sqrt[3]{24} \cong 2.88$, that is at least one of $\pm 1$ or $\pm 2$ must be a root.

Try the first few:

$f \left(1\right) = 1 - 3 - 10 + 24 = 12$
$f \left(- 1\right) = - 1 - 3 + 10 + 24 = 30$
$f \left(2\right) = 8 - 12 - 20 + 24 = 0$

So $x = 2$ is a root of $f \left(x\right) = 0$ and $\left(x - 2\right)$ is a factor of $f \left(x\right)$

Next use, synthetic division to divide $f \left(x\right)$ by $\left(x - 2\right)$...

Choose the first multiplier of $\left(x - 2\right)$ to match the leading term ${x}^{3}$. The multiplier we need is $\textcolor{red}{{x}^{2}}$

${x}^{2} \cdot \left(x - 2\right) = {x}^{3} - 2 {x}^{2}$

Subtract this from ${x}^{3} - 3 {x}^{2} - 10 x + 24$ to give a remainder:

$\left({x}^{3} - 3 {x}^{2} - 10 x + 24\right) - \left({x}^{3} - 2 {x}^{2}\right)$

$= - {x}^{2} - 10 x + 24$

Choose the next multiplier of $\left(x - 2\right)$ to match the leading term $- {x}^{2}$ of this remainder. The multiplier we need is $\textcolor{red}{- x}$

$- x \cdot \left(x - 2\right) = - {x}^{2} + 2 x$

Subtract this from the previous remainder to get a new remainder:

$\left(- {x}^{2} - 10 x + 24\right) - \left(- {x}^{2} + 2 x\right)$

$= - 12 x + 24$

Choose the next multiplier of $\left(x - 2\right)$ to match the leading term $- 12 x$ of this remainder. The multiplier we need is $\textcolor{red}{- 12}$

$- 12 \left(x - 2\right) = - 12 x + 24$.

Since this matches the previous remainder, we are done. All that remains to complete the division is to add the multipliers we found together to get:

${x}^{3} - 3 {x}^{2} - 10 x + 24 = \left(x - 2\right) \left({x}^{2} - x - 12\right)$

To factor $\left({x}^{2} - x - 12\right)$ we need a pair of factors of $12$ whose difference is $1$. The pair $4 , 3$ works.

Hence $\left(x - 4\right) \left(x + 3\right) = {x}^{2} - x - 12$

So, putting it all together:

${x}^{3} - 3 {x}^{2} - 10 x + 24 = \left(x - 2\right) \left(x - 4\right) \left(x + 3\right)$