# How do you use synthetic division to divide 2x^3-3x^2-5x-12div x-3?

Feb 25, 2016

$2 {x}^{2} + 3 x + 4$

#### Explanation:

The first thing we must do is find the value that makes $x - 3$ equal to zero. In this case, $x = 3$. This will be our divisor.

Now, we set up the problem:
If we have $\textcolor{g r e e n}{2} {x}^{3} \textcolor{red}{- 3} {x}^{2} \textcolor{b l u e}{- 5} x \textcolor{p u r p \le}{- 12}$

$\textcolor{w h i t e}{3.}$$|$$\textcolor{g r e e n}{2} \textcolor{w h i t e}{3.} \textcolor{red}{- 3} \textcolor{w h i t e}{3.} \textcolor{b l u e}{- 5} \textcolor{w h i t e}{3.} \textcolor{p u r p \le}{- 12}$
3$\textcolor{w h i t e}{.}$$|$______

Now, we bring down the $\textcolor{g r e e n}{2}$ and multiply it by the $3$, like this:
$\textcolor{w h i t e}{3.}$$|$$\textcolor{g r e e n}{2} \textcolor{w h i t e}{3.} \textcolor{red}{- 3} \textcolor{w h i t e}{3.} \textcolor{b l u e}{- 5} \textcolor{w h i t e}{3.} \textcolor{p u r p \le}{- 12}$
3$\textcolor{w h i t e}{.}$$|$__$6$____
$\textcolor{w h i t e}{\ldots \ldots .} \textcolor{g r e e n}{2}$

Now we add the $\textcolor{red}{- 3}$ to the $6$, which gives us $3$. I'll show you what I mean.

$\textcolor{w h i t e}{3.}$$|$$\textcolor{g r e e n}{2} \textcolor{w h i t e}{3.} \textcolor{red}{- 3} \textcolor{w h i t e}{3.} \textcolor{b l u e}{- 5} \textcolor{w h i t e}{3.} \textcolor{p u r p \le}{- 12}$
3$\textcolor{w h i t e}{.}$$|$__$6$___$9$_$12$__
$\textcolor{w h i t e}{\ldots \ldots .} \textcolor{g r e e n}{2}$$\textcolor{w h i t e}{\ldots .} 3$$\textcolor{w h i t e}{\ldots \ldots .} 4$$\textcolor{w h i t e}{\ldots \ldots .} 0$

I just did the whole thing, and I hope you can see what I did. I took the sum of the answer and multiplied it to the divisor, $3$. Then I take the product and place it in the next column. Then, I add the column together, and whatever the answer is, I multiply it to the $3$.

Anyways, we take the leftover number as the bottom, the $\textcolor{g r e e n}{2} \textcolor{w h i t e}{.} 3 \textcolor{w h i t e}{.} 4 \textcolor{w h i t e}{.} 0$ and rewrite them as an equation, like this: $2 {x}^{2} + 3 x + 4$. We can factor this further if we want, but I'm going to stop there.

Feb 25, 2016

$\left(2 {x}^{3} - 3 {x}^{2} - 5 x - 12\right) \div \left(x - 3\right) = 2 {x}^{2} + 3 x + 4$

#### Explanation:

We write the polynomial and its divisor down in long division form and work through the normal steps of long division:

Then we guess our first term in the quotient, which should subtract from the first term in the dividend:

After performing the subraction, we a left with our first remainder. We next drop the remaining terms in the dividend and guess our next term in the quotient:

and repeat for the last:

Which confirms that $x - 3$ is a factor of our polynomial evidenced by the fact that we get a zero remainder. So we conclude that:

$\left(2 {x}^{3} - 3 {x}^{2} - 5 x - 12\right) \div \left(x - 3\right) = 2 {x}^{2} + 3 x + 4$