Examples

\frac{6 x^{3} + 4 x^{2} - x}{x^{2} - 4}

$\frac{6 x^{3} + 4 x^{2} - x}{x^{2} - 4}$

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

$0$ .

x^{2}

$x^{2}$

0 x

$0 x$

4

$4$

6 x^{3}

$6 x^{3}$

4 x^{2}

$4 x^{2}$

x

$x$

0

$0$

Step 2

Divide the highest order term in the dividend

6 x^{3}

$6 x^{3}$ by the highest order term in divisor

x^{2}

$x^{2}$ .

6 x

$6 x$

x^{2}

$x^{2}$

0 x

$0 x$

4

$4$

6 x^{3}

$6 x^{3}$

4 x^{2}

$4 x^{2}$

x

$x$

0

$0$

Step 3

Multiply the new quotient term by the divisor.

6 x

$6 x$

x^{2}

$x^{2}$

0 x

$0 x$

4

$4$

6 x^{3}

$6 x^{3}$

4 x^{2}

$4 x^{2}$

x

$x$

0

$0$

6 x^{3}

$6 x^{3}$

0

$0$

24 x

$24 x$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in

6 x^{3} + 0 - 24 x

$6 x^{3} + 0 - 24 x$

6 x

$6 x$

x^{2}

$x^{2}$

0 x

$0 x$

4

$4$

6 x^{3}

$6 x^{3}$

4 x^{2}

$4 x^{2}$

x

$x$

0

$0$

6 x^{3}

$6 x^{3}$

0

$0$

24 x

$24 x$

Step 5

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

6 x

$6 x$

x^{2}

$x^{2}$

0 x

$0 x$

4

$4$

6 x^{3}

$6 x^{3}$

4 x^{2}

$4 x^{2}$

x

$x$

0

$0$

6 x^{3}

$6 x^{3}$

0

$0$

24 x

$24 x$

4 x^{2}

$4 x^{2}$

23 x

$23 x$

Step 6

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

6 x

$6 x$

x^{2}

$x^{2}$

0 x

$0 x$

4

$4$

6 x^{3}

$6 x^{3}$

4 x^{2}

$4 x^{2}$

x

$x$

0

$0$

6 x^{3}

$6 x^{3}$

0

$0$

24 x

$24 x$

4 x^{2}

$4 x^{2}$

23 x

$23 x$

0

$0$

Step 7

Divide the highest order term in the dividend

4 x^{2}

$4 x^{2}$ by the highest order term in divisor

x^{2}

$x^{2}$ .

6 x

$6 x$

4

$4$

x^{2}

$x^{2}$

0 x

$0 x$

4

$4$

6 x^{3}

$6 x^{3}$

4 x^{2}

$4 x^{2}$

x

$x$

0

$0$

6 x^{3}

$6 x^{3}$

0

$0$

24 x

$24 x$

4 x^{2}

$4 x^{2}$

23 x

$23 x$

0

$0$

Step 8

Multiply the new quotient term by the divisor.

6 x

$6 x$

4

$4$

x^{2}

$x^{2}$

0 x

$0 x$

4

$4$

6 x^{3}

$6 x^{3}$

4 x^{2}

$4 x^{2}$

x

$x$

0

$0$

6 x^{3}

$6 x^{3}$

0

$0$

24 x

$24 x$

4 x^{2}

$4 x^{2}$

23 x

$23 x$

0

$0$

4 x^{2}

$4 x^{2}$

0

$0$

16

$16$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in

4 x^{2} + 0 - 16

$4 x^{2} + 0 - 16$

6 x

$6 x$

4

$4$

x^{2}

$x^{2}$

0 x

$0 x$

4

$4$

6 x^{3}

$6 x^{3}$

4 x^{2}

$4 x^{2}$

x

$x$

0

$0$

6 x^{3}

$6 x^{3}$

0

$0$

24 x

$24 x$

4 x^{2}

$4 x^{2}$

23 x

$23 x$

0

$0$

4 x^{2}

$4 x^{2}$

0

$0$

16

$16$

Step 10

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

6 x

$6 x$

4

$4$

x^{2}

$x^{2}$

0 x

$0 x$

4

$4$

6 x^{3}

$6 x^{3}$

4 x^{2}

$4 x^{2}$

x

$x$

0

$0$

6 x^{3}

$6 x^{3}$

0

$0$

24 x

$24 x$

4 x^{2}

$4 x^{2}$

23 x

$23 x$

0

$0$

4 x^{2}

$4 x^{2}$

0

$0$

16

$16$

23 x

$23 x$

16

$16$

Step 11

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

6 x + 4 + \frac{23 x + 16}{x^{2} - 4}

$6 x + 4 + \frac{23 x + 16}{x^{2} - 4}$

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