Question

select the correct answer.
using synthetic division, find $(2x^{4}+4x^{3}+2x^{2}+8x+8)\div(x+2)$.
a. $2x^{3}+2x+4$
b. $2x^{4}+2x^{2}+4x$
c. $2x^{3}+2x+4+\frac{1}{x+2}$
d. $2x^{3}+2x^{2}+4$

Explanation:

Step1: Identify root of divisor

For $x+2=0$, root is $x=-2$.
Coefficients of dividend: $2, 4, 2, 8, 8$

Step2: Set up synthetic division

Write root and coefficients:

$$\begin{array}{r|rrrrr} -2 & 2 & 4 & 2 & 8 & 8 \\ \hline & & -4 & 0 & -4 & -8 \\ \hline & 2 & 0 & 2 & 4 & 0 \end{array}$$

Step3: Interpret results

Last value is remainder (0).
Quotient coefficients: $2,0,2,4$, so quotient is $2x^3+0x^2+2x+4=2x^3+2x+4$.

Answer:

A. $2x^3 + 2x + 4$