Question

divide by using synthetic division. simplify your answer as much as possible.
$\frac{3x^{3}+x^{2}+7x+2}{3x-2} = \square$

Explanation:

Step1: Identify root of divisor

For $3x-2=0$, solve for $x$: $x=\frac{2}{3}$

Step2: Set up synthetic division

Coefficients of dividend: $3, 1, 7, 2$; root: $\frac{2}{3}$

$$\begin{array}{r|rrrr} \frac{2}{3} & 3 & 1 & 7 & 2 \\ & & 2 & 2 & 6 \\ \hline & 3 & 3 & 9 & 8 \\ \end{array}$$

Step3: Write quotient and remainder

Quotient: $3x^2+3x+9$, remainder: $8$. Divide quotient by 3 (from divisor $3x-2$):
Quotient becomes $x^2+x+3$, remainder stays $8$.

Step4: Express final result

Form: $\text{Quotient} + \frac{\text{Remainder}}{\text{Divisor}}$

Answer:

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