# How do you divide (4x^5 +8x^4 -8x^3 +4x^2 +x-8) / (5x^2 -4x+9)?

Mar 23, 2016

Since coefficient of highest power of $x$, i.e ., ${x}^{2}$ term in the denominator is $\ne 1$, therefore we need to divide using the long division method.
Quotient $= \frac{4}{5} {x}^{3} + \frac{56}{25} {x}^{2} - \frac{156}{125} x + \frac{1396}{625}$
Remainder$= \frac{13229}{625} x - \frac{17564}{625}$

#### Explanation:

$\textcolor{w h i t e}{W W W W W W W} \frac{4}{5} {x}^{3} + \frac{56}{25} {x}^{2} - \frac{156}{125} x + \frac{1396}{625}$
5x^2-4x+9)bar(4x^5+8x^4-8x^3+4x^2+x-8)(
$\textcolor{w h i t e}{W W W W W W} 4 {x}^{5} - \frac{16}{5} {x}^{4} + \frac{36}{5} {x}^{3}$
$\textcolor{w h i t e}{W W W W W} \underline{- \textcolor{w h i t e}{i i W} + \textcolor{w h i t e}{W i W} - \textcolor{w h i t e}{W W W W W}}$
$\textcolor{w h i t e}{W W W W W W W W W} \frac{56}{5} {x}^{4} - \frac{76}{5} {x}^{3} + \text{ } 4 {x}^{2}$
$\textcolor{w h i t e}{W W W W W W W W W} \frac{56}{5} {x}^{4} - \frac{224}{25} {x}^{3} + \frac{504}{25} {x}^{2}$
$\textcolor{w h i t e}{W W W W W W W W} \underline{- \textcolor{w h i t e}{W i i W} + \textcolor{w h i t e}{W i W} - \textcolor{w h i t e}{W W W}}$
$\textcolor{w h i t e}{W W W W W W W W W W W} - \frac{156}{25} {x}^{3} + \frac{404}{25} {x}^{2} + x$
$\textcolor{w h i t e}{W W W W W W W W W W W} - \frac{156}{25} {x}^{3} + \frac{624}{125} {x}^{2} - \frac{1404}{125} x$
$\textcolor{w h i t e}{W W W W W W I W W} \underline{\textcolor{w h i t e}{W i i W} + \textcolor{w h i t e}{W W W} - \textcolor{w h i t e}{i W W} +}$
$\textcolor{w h i t e}{W W W W W W W W W W W W W} \frac{1396}{125} {x}^{2} + \frac{1529}{125} x - 8$
$\textcolor{w h i t e}{W W W W W W W W W W W W W} \frac{1396}{125} {x}^{2} - \frac{5584}{625} x + \frac{12564}{625}$
$\textcolor{w h i t e}{W W W W W W W i W W} \underline{\textcolor{w h i t e}{W i i W} - \textcolor{w h i t e}{W W W} + \textcolor{w h i t e}{W i W W} -}$
$\textcolor{w h i t e}{W W W W W W W W W W W W W W W W W} \frac{13229}{625} x - \frac{17564}{625}$