# What is (8x^3 + 12x^2 - 6x + 8)/(2x+6)?

Mar 31, 2017

$\frac{8 {x}^{3} + 12 {x}^{2} - 6 x + 8}{2 x + 6} = \left(4 {x}^{2} - 6 x + 15\right) - \frac{41}{x + 3}$
or
quotient $= \left(4 {x}^{2} - 6 x + 15\right)$ & remainder $= - 82$

#### Explanation:

$\frac{8 {x}^{3} + 12 {x}^{2} - 6 x + 8}{2 x + 6}$

dividing numerator and denominator by 2 (purely for simplification purpose)

= $\frac{4 {x}^{3} + 6 {x}^{2} - 3 x + 4}{x + 3}$

try to break the polynomial in the numerator such that we can factor out the denominator from successive terms

= $\frac{4 {x}^{3} + 12 {x}^{2} - 6 {x}^{2} - 18 x + 15 x + 45 - 41}{x + 3}$2

=$\frac{4 {x}^{2} \left(x + 3\right) - 6 x \left(x + 3\right) + 15 \left(x + 3\right) - 41}{x + 3}$

=$\frac{4 {x}^{2} \left(x + 3\right) - 6 x \left(x + 3\right) + 15 \left(x + 3\right)}{x + 3} - \frac{41}{x + 3}$

= $\frac{\left(x + 3\right) \left(4 {x}^{2} - 6 x + 15\right)}{x + 3} - \frac{41}{x + 3}$

=$\left(4 {x}^{2} - 6 x + 15\right) - \frac{41}{x + 3}$

the second term in the expression $i . e . - \frac{41}{x + 3}$ cannot be simplified further as degree of numerator is less than that of denominator.
[Degree is the highest power of variable in a polynomial
($\therefore$ degree of $- 41$ is $0$ and that of $x + 3$ is $1$ ) ]

$\therefore$ $\frac{8 {x}^{3} + 12 {x}^{2} - 6 x + 8}{2 x + 6} = \left(4 {x}^{2} - 6 x + 15\right) - \frac{41}{x + 3}$
or
quotient $= \left(4 {x}^{2} - 6 x + 15\right)$ & remainder $= 2 \cdot \left(- 41\right) = - 82$ since we had initially divided both numerator and denominator by 2, we have to multiply -41 by 2 to get the correct remainder.

Mar 31, 2017

color(blue)(4x^2-6x+15

#### Explanation:

$\textcolor{w h i t e}{a a a a a a a a a a}$$4 {x}^{2} - 6 x + 15$
$\textcolor{w h i t e}{a a a a a a a a a a}$$- - - - -$
$\textcolor{w h i t e}{a a a a} 2 x + 6$$|$$8 {x}^{3} + 12 {x}^{2} - 6 x + 8$color(white) (aaaa)∣$\textcolor{b l u e}{4 {x}^{2} - 6 x + 15}$
$\textcolor{w h i t e}{a a a a a a a a a a a}$$8 {x}^{3} + 24 {x}^{2}$color(white)
$\textcolor{w h i t e}{a a a a a a a a a a a}$$- - - -$
$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$0 - 12 {x}^{2} - 6 x$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a a}$$- 12 {x}^{2} - 36 x$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a}$$- - - - - -$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a a}$$30 x + 8$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a a}$$30 x + 90$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a a a}$$- - -$
$\textcolor{w h i t e}{a a a a a a a a a a a a a a a a a a a a a a a a a a a}$$- 82$

The remainder is =color(blue)(-82 and the quotient is =color(blue)(4x^2-6x+15