How do you use synthetic division to divide (4x^2 - 2x + 6) by 2x-3?

1 Answer
Oct 20, 2015

$\left(4 {x}^{2} - 2 x + 6\right) \div \left(2 x - 3\right) = \left(2 x + 2\right) R : 12$

Explanation:

$\left(\textcolor{b r o w n}{4} {x}^{2} \textcolor{b r o w n}{- 2} x \textcolor{b r o w n}{+ 6}\right) \div \left(\textcolor{c y a n}{2} x \textcolor{b l u e}{- 3}\right)$

Remember to reverse the sign on $\textcolor{b l u e}{\left(- 3\right)}$
and since we have a non-monic divisor, we need to divide each column sum by (in this case) $\textcolor{c y a n}{2}$
"Bring down" the first coefficient
Then divide by 2
{: ( ," | ",color(brown)(4),color(brown)(-2),color(brown)(+6)), (color(blue)(+3)," | ", , , ), ( ," | ","----","----","----"), (color(cyan)(/2)," | ",4,,), (," | ",2,color(white)("X")2,) :}

Multiply the last column quotient ($2$) by $3$ and write in the next column.
Add that column.
{: ( ," | ",4,-2,+6), (+3," | ", , color(white)("X")6 , ), ( ," | ","----","----","----"), (/2," | ",4,color(white)("X")4,), (," | ",2,,) :}

Repeat this process until done
{: ( ," | ",4,-2,+6), (+3," | ", , color(white)("X")6 , color(white)("X")6), ( ," | ","----","----","----"), (/2," | ",4,color(white)("X")4,color(white)("X")color(red)(12)), (," | ",color(green)(2),color(white)("X")color(orange)(2),) :}

The last sum (undivided), $\textcolor{red}{12}$, is the remainder.
The quotients preceding the last column, $\textcolor{g r e e n}{2}$ and $\textcolor{\mathmr{and} a n \ge}{2}$, are the coefficients of the quotient expression.

That is the solution is
$\textcolor{g r e e n}{2} x + \textcolor{\mathmr{and} a n \ge}{2}$ with a Remainder of $\textcolor{red}{12}$