Question

use the long division method to find the result when $9x^3 - 3x^2 + 9x + 3$ is divided by $3x + 2$. if there is a remainder, express the result in the form $q(x) + \frac{r(x)}{b(x)}$.
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Explanation:

Step1: Divide leading terms

$\frac{9x^3}{3x} = 3x^2$

Step2: Multiply divisor by result

$3x^2(3x+2) = 9x^3 + 6x^2$

Step3: Subtract from dividend

$(9x^3-3x^2+9x+3)-(9x^3+6x^2) = -9x^2+9x+3$

Step4: Divide new leading terms

$\frac{-9x^2}{3x} = -3x$

Step5: Multiply divisor by result

$-3x(3x+2) = -9x^2-6x$

Step6: Subtract from current polynomial

$(-9x^2+9x+3)-(-9x^2-6x) = 15x+3$

Step7: Divide new leading terms

$\frac{15x}{3x} = 5$

Step8: Multiply divisor by result

$5(3x+2) = 15x+10$

Step9: Subtract to find remainder

$(15x+3)-(15x+10) = -7$

Answer:

$3x^2 - 3x + 5 + \frac{-7}{3x+2}$