Question

use synthetic division to simplify $\frac{x^{5}+11x^{4}+36x^{3}+24x^{2}-32x + 1}{x + 5}$. 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 $x + 5$, so we use $- 5$ for synthetic - division. The coefficients of the dividend $x^{5}+11x^{4}+36x^{3}+24x^{2}-32x + 1$ are $1,11,36,24,-32,1$.

Step2: Bring down the first coefficient

Bring down the first coefficient $1$.

Step3: Multiply and add

Multiply $-5\times1=-5$, add to the second coefficient: $11+( - 5)=6$.

Step4: Repeat multiplication and addition

Multiply $-5\times6=-30$, add to the third coefficient: $36+( - 30)=6$.
Multiply $-5\times6=-30$, add to the fourth coefficient: $24+( - 30)=-6$.
Multiply $-5\times(-6) = 30$, add to the fifth coefficient: $-32 + 30=-2$.
Multiply $-5\times(-2)=10$, add to the sixth coefficient: $1+10 = 11$.

Step5: Write the quotient and remainder

The quotient $q(x)=x^{4}+6x^{3}+6x^{2}-6x - 2$ and the remainder $r = 11$, and $d(x)=x + 5$.

Answer:

$x^{4}+6x^{3}+6x^{2}-6x - 2+\frac{11}{x + 5}$