Divide (x^3+2x^2)/(3x^2-1)

$\frac{x^{3} + 2 x^{2}}{3 x^{2} - 1}$

Step 1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$3 x^{2}$

$0 x$

$1$

$x^{3}$

$2 x^{2}$

$0 x$

$0$

Step 2

Divide the highest order term in the dividend $x^{3}$ by the highest order term in divisor $3 x^{2}$ .

$\frac{x}{3}$

$3 x^{2}$

$0 x$

$1$

$x^{3}$

$2 x^{2}$

$0 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$\frac{x}{3}$

$3 x^{2}$

$0 x$

$1$

$x^{3}$

$2 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$\frac{x}{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{3} + 0 - \frac{x}{3}$

$\frac{x}{3}$

$3 x^{2}$

$0 x$

$1$

$x^{3}$

$2 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$\frac{x}{3}$

Step 5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$\frac{x}{3}$

$3 x^{2}$

$0 x$

$1$

$x^{3}$

$2 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$\frac{x}{3}$

$2 x^{2}$

$\frac{x}{3}$

Step 6

Pull the next terms from the original dividend down into the current dividend.

$\frac{x}{3}$

$3 x^{2}$

$0 x$

$1$

$x^{3}$

$2 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$\frac{x}{3}$

$2 x^{2}$

$\frac{x}{3}$

$0$

Step 7

Divide the highest order term in the dividend $2 x^{2}$ by the highest order term in divisor $3 x^{2}$ .

$\frac{x}{3}$

$\frac{2}{3}$

$3 x^{2}$

$0 x$

$1$

$x^{3}$

$2 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$\frac{x}{3}$

$2 x^{2}$

$\frac{x}{3}$

$0$

Step 8

Multiply the new quotient term by the divisor.

$\frac{x}{3}$

$\frac{2}{3}$

$3 x^{2}$

$0 x$

$1$

$x^{3}$

$2 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$\frac{x}{3}$

$2 x^{2}$

$\frac{x}{3}$

$0$

$2 x^{2}$

$0$

$\frac{2}{3}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} + 0 - \frac{2}{3}$

$\frac{x}{3}$

$\frac{2}{3}$

$3 x^{2}$

$0 x$

$1$

$x^{3}$

$2 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$\frac{x}{3}$

$2 x^{2}$

$\frac{x}{3}$

$0$

$2 x^{2}$

$0$

$\frac{2}{3}$

Step 10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

$\frac{x}{3}$

$\frac{2}{3}$

$3 x^{2}$

$0 x$

$1$

$x^{3}$

$2 x^{2}$

$0 x$

$0$

$x^{3}$

$0$

$\frac{x}{3}$

$2 x^{2}$

$\frac{x}{3}$

$0$

$2 x^{2}$

$0$

$\frac{2}{3}$

$\frac{x}{3}$

$\frac{2}{3}$

Step 11

The final answer is the quotient plus the remainder over the divisor.

$\frac{x}{3} + \frac{2}{3} + \frac{\frac{x}{3} + \frac{2}{3}}{3 x^{2} - 1}$

Divide (x^3+2x^2)/(3x^2-1)

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