What is the quotient of (x^3+3x^2-3x-12)div(x^2+5x+6)?

Apr 20, 2017

$x - 2$

Explanation:

One way of dividing a rational function is to use the divisor as a factor in the numerator.

$\text{Consider the numerator}$

$\textcolor{red}{x} \left({x}^{2} + 5 x + 6\right) \textcolor{m a \ge n t a}{- 5 {x}^{2} - 6 x} + 3 {x}^{2} - 3 x - 12$

$= \textcolor{red}{x} \left({x}^{2} + 5 x + 6\right) \textcolor{red}{- 2} \left({x}^{2} + 5 x + 6\right) \textcolor{m a \ge n t a}{+ 10 x + 12} - 9 x - 12$

$= \textcolor{red}{x} \left({x}^{2} + 5 x + 6\right) \textcolor{red}{- 2} \left({x}^{2} + 5 x + 6\right) + x$

$\text{quotient is " color(red)(x-2)" and remainder is } x$

$\Rightarrow \frac{{x}^{3} + 3 {x}^{2} - 3 x - 12}{{x}^{2} + 5 x + 6} = x - 2 + \frac{x}{{x}^{2} + 5 x + 6}$