Question

use synthetic division to simplify $\frac{8x^{4}-31x^{3}-4x^{2}}{x - 4}$. write your answer in the form $q(x)+\frac{r}{d(x)}$, where $q(x)$ is a polynomial, $r$ is an integer, and $d(x)$ is a linear polynomial. simplify any fractions.

Explanation:

Step1: Set up synthetic division

The divisor is $x - 4$, so we use $c = 4$. The coefficients of the dividend $8x^{4}-31x^{3}-4x^{2}+0x + 0$ are $8,-31,-4,0,0$.

Step2: Bring down the first coefficient

Bring down the first coefficient $8$.

Step3: Multiply and add

Multiply $4\times8 = 32$, add to $- 31$: $-31 + 32=1$. Then multiply $4\times1 = 4$, add to $-4$: $-4 + 4 = 0$. Multiply $4\times0 = 0$, add to $0$: $0+0 = 0$. Multiply $4\times0 = 0$, add to $0$: $0 + 0=0$.

Step4: Write the quotient and remainder

The quotient polynomial $q(x)=8x^{3}+x^{2}$ and the remainder $r = 0$, and $d(x)=x - 4$.

Answer:

$8x^{3}+x^{2}+\frac{0}{x - 4}=8x^{3}+x^{2}$