Question

use synthetic division to simplify $\frac{7x^{5}+22x^{4}-19x^{2}-20x + 9}{x + 3}$. 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 $x + 3$, so we use $- 3$ for synthetic - division. The coefficients of the dividend $7x^{5}+22x^{4}+0x^{3}-19x^{2}-20x + 9$ are $7,22,0,-19,-20,9$.

Step2: Bring down the first coefficient

Bring down the first coefficient $7$.

Step3: Multiply and add

Multiply $-3$ by $7$ to get $-21$, and add to $22$: $22+( - 21)=1$.

Step4: Repeat multiplication and addition

Multiply $-3$ by $1$ to get $-3$, and add to $0$: $0+( - 3)=-3$. Multiply $-3$ by $-3$ to get $9$, and add to $-19$: $-19 + 9=-10$. Multiply $-3$ by $-10$ to get $30$, and add to $-20$: $-20 + 30 = 10$. Multiply $-3$ by $10$ to get $-30$, and add to $9$: $9+( - 30)=-21$.

Step5: Write the quotient and remainder

The quotient polynomial $q(x)$ has coefficients $7,1,-3,-10,10$ and degree one less than the dividend, so $q(x)=7x^{4}+x^{3}-3x^{2}-10x + 10$, and the remainder $r=-21$, and the divisor $d(x)=x + 3$.

Answer:

$7x^{4}+x^{3}-3x^{2}-10x + 10-\frac{21}{x + 3}$