Question

use synthetic division to find $(x^{3}-4x + 33)div(x + 3)$. 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: Identify root of divisor

For $x+3=0$, root is $x=-3$.
List coefficients of dividend $x^3 + 0x^2 -4x +33$: $1, 0, -4, 33$

Step2: Set up synthetic division

Bring down the leading coefficient:
$1$

Multiply by $-3$: $1\times(-3) = -3$
Add to next coefficient: $0 + (-3) = -3$

Step3: Continue synthetic division

Multiply $-3$ by $-3$: $9$
Add to next coefficient: $-4 + 9 = 5$

Step4: Final synthetic division step

Multiply $5$ by $-3$: $-15$
Add to last coefficient: $33 + (-15) = 18$

Step5: Form quotient and remainder

Coefficients of quotient: $1, -3, 5$ → $q(x)=x^2-3x+5$
Remainder $r=18$, divisor $d(x)=x+3$

Answer:

$x^2 - 3x + 5 + \frac{18}{x+3}$