Question

divide using long division. state the quotient, q(x), and the remainder, r(x).
\frac{2x^{3}-5x^{2}+2x + 8}{2x^{2}+x + 7}
\frac{2x^{3}-5x^{2}+2x + 8}{2x^{2}+x + 7}=\square+\frac{\square}{2x^{2}+x + 7}
(simplify your answers. do not factor. use integers or fractions for any numbers in the express)

Explanation:

Step1: Divide leading terms

Divide $2x^{3}$ by $2x^{2}$ to get $x$.

Step2: Multiply and subtract

Multiply $2x^{2}+x + 7$ by $x$: $x(2x^{2}+x + 7)=2x^{3}+x^{2}+7x$. Subtract from $2x^{3}-5x^{2}+2x + 8$: $(2x^{3}-5x^{2}+2x + 8)-(2x^{3}+x^{2}+7x)=-6x^{2}-5x + 8$.

Step3: Divide new leading term

Divide $-6x^{2}$ by $2x^{2}$ to get $- 3$.

Step4: Multiply and subtract again

Multiply $2x^{2}+x + 7$ by $-3$: $-3(2x^{2}+x + 7)=-6x^{2}-3x-21$. Subtract from $-6x^{2}-5x + 8$: $(-6x^{2}-5x + 8)-(-6x^{2}-3x-21)=-2x + 29$.

Answer:

$q(x)=x - 3$, $r(x)=-2x + 29$, $\frac{2x^{3}-5x^{2}+2x + 8}{2x^{2}+x + 7}=x - 3+\frac{-2x + 29}{2x^{2}+x + 7}$