synthetic divide polynomial $(2x^3+7x^2-13x-3) \div (2x-3)$ I need to figure out how synthetic division works.  The problem is $$ (2x^3+7x^2-13X-3) \div (2x-3)$$ I can do the long division.
$$\require{enclose}
\begin{array}{r}
x^2+5x+1 \\
      (2x-3) \enclose{longdiv}{2x^3+7x^2-13x-3}\\ \underline{-2x^3+3x^2} \phantom{100000000} \\ 10x-13x \phantom{1000} \\\underline{-10x+15x} \phantom{1000} \\ 2x-3 \\\underline{-2x+3} \\ 0
\end{array}$$
But when I do synthetic division my answer is slightly different;
$$ 3/2 \left[ \begin{array}{r}{2 \phantom{10}7 \phantom{2}-13\phantom{2}-3} \\ \underline{ \phantom{100}3 \phantom{100} 15 \phantom{1000} 3} \\ 2 \phantom{1}10\phantom{1000} 2 \phantom{1000} 0\end{array}  \right] = 2x^2 +10x +2$$
I just realized that I could factor out the two from the polynomial after I synthetic divide.  The answer then would be; $$2(x^2+5x+1)$$  The only problem I have is the two graphs would be different.  The second graph would stretch along the y axis.  If I put a 1 in for x then y =14 a
fter synthetic division.  Y would  = 7  if  I put a 1 in for x after I use long division.   I am not sure what to do next.
 A: When you do synthetic division, you're not dividing by $2x-3$, you're dividing by $x - 3/2$.
Note that $2(x-3/2)  = 2x-3$.
A: Since you have taken $3/2$ while doing synthetic division, we can write
$$ 2x^3 +7x^2-13x -3 = (x- \frac32)(2x^2 + 10x + 2) $$
Now rearranging terms in RHS we get
$$2x^3 +7x^2-13x -3 = (2x- 3)(x^2 + 5x + 1)$$ which is the same answer you got when  you did the long division.
A: When I was writing the question I was hoping to get the steps to synthetic division.  I thought I'd show the steps that I came up with from the discussion.
Step 1: is to get the divisor in (x-a) format.  There cannot be a coefficient.  It is done by factoring:
$$(2x-3)=2(x- \frac32)$$
The problem now looks like this $$\frac{2x^3+7x^2-13x-3}{2(x-\frac32)}$$
Step 2: is divide by 2 $$ \frac{\frac22x^3+ \frac72x^2 - \frac{13}{2}x-\frac32}{x- \frac32}$$
Step 3: Now the problem is setup for synthetic division
$$  \frac32 \left[ \begin{array}{l}1 \phantom{10}3.5 -6.5 -1.5 \\ \underline{\phantom{100}1.5 \phantom{100}7.5 \phantom{10}1.5} \\1 \phantom{100}5 \phantom{100}1 \phantom{1000}0\end{array}\right] = x^2 +5x+1$$
