# How do you use synthetic division to divide (2x^2 – 23x + 63) ÷ (x – 7)?

##### 1 Answer
Dec 8, 2015

(see below for details of synthetic division method)
2x^2-23x+63)div (x-7) = (2x-9) R: 0

#### Explanation:

Set up for synthetic division by monic binomial divisor:

row : the coefficients of the terms of the dividend polynomial (in descending exponent sequence)

row : the negative of the constant from the divisor (the $\times$ sign may be omitted; it is simply there as a reminder for later processing)
and
a copy of the first dividend coefficient

{: (,"|",2,-23,color(white)("X")+6,color(white)("XXXX")"row "), (,"|",,,,color(white)("XXXX")"row "), ("-----",,"-----","-----","-----",), (xx (+7),"|",2,,,color(white)("XXXX")"row ") :}

$\text{--------------------------------------------------------------------------------}$

Repeat until you have entered a value in the last column of row 

Multiply the last value entered in row  by the negative of the divisor coefficient and write the product in the next column of row .

Add the values in rows  and  to get the next column value for row 
{: (,"|",2,-23,color(white)("X")+6,color(white)("XXXX")"row "), (,"|",,color(white)("X")14,,color(white)("XXXX")"row "), ("-----",,"-----","-----","-----",), (xx (+7),"|",2,-9,,color(white)("XXXX")"row ") :}

$\text{--------------------------------------------------------------------------------}$

{: (,"|",2,-23,color(white)("X")+6,color(white)("XXXX")"row "), (,"|",,color(white)("X")14,-63,color(white)("XXXX")"row "), ("-----",,"-----","-----","-----",), (xx (+7),"|",color(blue)(2),color(blue)(-9),color(white)("XX")color(red)(0),color(white)("XXXX")"row ") :}

$\text{--------------------------------------------------------------------------------}$

The $\textcolor{red}{\text{last sum}}$ is the Remainder
The $\textcolor{b l u e}{\text{preceding sums}}$ are the coefficients of the reduced quotient.

i.e. the final answer is $\textcolor{b l u e}{2} x \textcolor{b l u e}{- 9}$ with a remainder of $\textcolor{red}{0}$