How do you divide x^3+7x^2-3x+4) div(x+2) and identify any restrictions on the variable?

Dec 20, 2017

${x}^{2} + 5 x - 13 + \frac{30}{x + 2} \to \left(x \ne - 2\right)$

Explanation:

$\text{one way is to use the divisor as a factor in the numerator}$

$\text{consider the numerator}$

$\textcolor{red}{{x}^{2}} \left(x + 2\right) \textcolor{m a \ge n t a}{- 2 {x}^{2}} + 7 {x}^{2} - 3 x + 4$

$= \textcolor{red}{{x}^{2}} \left(x + 2\right) \textcolor{red}{+ 5 x} \left(x + 2\right) \textcolor{m a \ge n t a}{- 10 x} - 3 x + 4$

$= \textcolor{red}{{x}^{2}} \left(x + 2\right) \textcolor{red}{+ 5 x} \left(x + 2\right) \textcolor{red}{- 13} \left(x + 2\right) \textcolor{m a \ge n t a}{+ 26} + 4$

$= \textcolor{red}{{x}^{2}} \left(x + 2\right) \textcolor{red}{+ 5 x} \left(x + 2\right) \textcolor{red}{- 13} \left(x + 2\right) + 30$

$\text{quotient "=color(red)(x^2+5x-13)," remainder } = 30$

$\Rightarrow \frac{{x}^{3} + 7 {x}^{2} - 3 x + 4}{x + 2}$

$= {x}^{2} + 5 x - 13 + \frac{30}{x + 2} \to \left(x \ne - 2\right)$