How do you divide ( x^4-2x^3-2x^2+11x+12)/(2x-6)?

Jan 7, 2016

Long divide coefficients to find:

$\frac{{x}^{4} - 2 {x}^{3} - 2 {x}^{2} + 11 x + 12}{2 x - 6} = \frac{1}{2} {x}^{3} + \frac{1}{2} {x}^{2} + \frac{1}{2} x + 7$

with remainder $27$

Explanation:

I like to long divide coefficients, but first note that $2 x - 6 = 2 \left(x - 3\right)$, so to simplify the calculation, let's divide by $x - 3$, then divide the result (and any remainder) by $2$ at the end:

We find:

$\frac{{x}^{4} - 2 {x}^{3} - 2 {x}^{2} + 11 x + 12}{x - 3} = {x}^{3} + {x}^{2} + x + 14$

with remainder $54$.

Hence (dividing by $2$):

$\frac{{x}^{4} - 2 {x}^{3} - 2 {x}^{2} + 11 x + 12}{2 x - 6} = \frac{1}{2} {x}^{3} + \frac{1}{2} {x}^{2} + \frac{1}{2} x + 7$

with remainder $27$