# I need to figure out how synthetic division works. The problem is ( 2 x 3 </msup>

I need to figure out how synthetic division works. The problem is
$\left(2{x}^{3}+7{x}^{2}-13x-3\right)÷\left(2x-3\right)$
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fugprurgeil
Since you have taken $3/2$ while doing synthetic division, we can write
$2{x}^{3}+7{x}^{2}-13x-3=\left(x-\frac{3}{2}\right)\left(2{x}^{2}+10x+2\right)$
Now rearranging terms in RHS we get
$2{x}^{3}+7{x}^{2}-13x-3=\left(2x-3\right)\left({x}^{2}+5x+1\right)$
which is the same answer you got when you did the long division.

antennense
Step 1: is to get the divisor in (x-a) format. There cannot be a coefficient. It is done by factoring:
$\left(2x-3\right)=2\left(x-\frac{3}{2}\right)$
The problem now looks like this
$\frac{2{x}^{3}+7{x}^{2}-13x-3}{2\left(x-\frac{3}{2}\right)}$
Step 2: is divide by 2
$\frac{\frac{2}{2}{x}^{3}+\frac{7}{2}{x}^{2}-\frac{13}{2}x-\frac{3}{2}}{x-\frac{3}{2}}$
Step 3: Now the problem is setup for synthetic division
$\frac{3}{2}\left[\begin{array}{l}1\phantom{10}3.5-6.5-1.5\\ \underset{_}{\phantom{100}1.5\phantom{100}7.5\phantom{10}1.5}\\ 1\phantom{100}5\phantom{100}1\phantom{1000}0\end{array}\right]={x}^{2}+5x+1$