Question

Using synthetic division find the quotient and remainder of x^3 + 2x^2-2x +4 divided by x+5

Answer 1

We want to calculate the following division:

$\frac{x^3+2x^2-2x+4}{x+5}$

We start by dividing the leading term of the dividend by the leading term of the divisor:

$\frac{x^3}{x}=x^2$

Then, we multiply this result by the divisor:

$x^2(x+5)=x^3+5x^2$

Subtract the dividend from the obtained result:

$(x^3+2x^2-2x+4)-(x^3+5x^2)=-3x^2-2x+4$

Then, we have:

$\frac{x^{3}+2x^{2}-2x+4}{x+5}=x^2+\frac{-3x^2-2x+4}{x+5}$

Since the second term can still be divided(the degree of the numerator polynomial is bigger than the degree of our divisor), we repeat the process:

$\begin{gathered} \frac{-3x^2}{x}=-3x \\ \\ -3x(x+5)=-3x^2-15x \\ \\ (-3x^2-2x+4)-(-3x^2-15x)=13x+4 \\ \\ x^2+\frac{-3x^2-2x+4}{x+5}=x^2-3x+\frac{13x+4}{x+5} \\ \\ \frac{13x}{x}=13 \\ \\ 13(x+5)=13x+65 \\ \\ (13x+4)-(13x+65)=-61 \\ \\ x^2-3x+\frac{13x+4}{x+5}=x^2-3x+13-\frac{61}{x+5} \end{gathered}$

And this is our result:

$\frac{x^{3}+2x^{2}-2x+4}{x+5}=x^2-3x+13-\frac{61}{x+5}$