Algebra Examples

$\frac{2 x^{4} + 8 x^{3} - 25 x^{2} - 6 x + 14}{x + 6}$

Step 1

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

$x$

$6$

$2 x^{4}$

$8 x^{3}$

$25 x^{2}$

$6 x$

$14$

Step 2

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

$2 x^{3}$

$x$

$6$

$2 x^{4}$

$8 x^{3}$

$25 x^{2}$

$6 x$

$14$

Step 3

Multiply the new quotient term by the divisor.

$2 x^{3}$

$x$

$6$

$2 x^{4}$

$8 x^{3}$

$25 x^{2}$

$6 x$

$14$

$2 x^{4}$

$12 x^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{4} + 12 x^{3}$

$2 x^{3}$

$x$

$6$

$2 x^{4}$

$8 x^{3}$

$25 x^{2}$

$6 x$

$14$

$2 x^{4}$

$12 x^{3}$

Step 5

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

$2 x^{3}$

$x$

$6$

$2 x^{4}$

$8 x^{3}$

$25 x^{2}$

$6 x$

$14$

$2 x^{4}$

$12 x^{3}$

$4 x^{3}$

Step 6

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

$2 x^{3}$

$x$

$6$

$2 x^{4}$

$8 x^{3}$

$25 x^{2}$

$6 x$

$14$

$2 x^{4}$

$12 x^{3}$

$4 x^{3}$

$25 x^{2}$

Step 7

Divide the highest order term in the dividend $- 4 x^{3}$ by the highest order term in divisor $x$ .

$2 x^{3}$

$4 x^{2}$

$x$

$6$

$2 x^{4}$

$8 x^{3}$

$25 x^{2}$

$6 x$

$14$

$2 x^{4}$

$12 x^{3}$

$4 x^{3}$

$25 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

				$2 x^{3}$	-	$4 x^{2}$
$x$	+	$6$		$2 x^{4}$	+	$8 x^{3}$	-	$25 x^{2}$	-	$6 x$	+	$14$
			-	$2 x^{4}$	-	$12 x^{3}$
					-	$4 x^{3}$	-	$25 x^{2}$
					-	$4 x^{3}$	-	$24 x^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{3} - 24 x^{2}$

$2 x^{3}$

$4 x^{2}$

$x$

$6$

$2 x^{4}$

$8 x^{3}$

$25 x^{2}$

$6 x$

$14$

$2 x^{4}$

$12 x^{3}$

$4 x^{3}$

$25 x^{2}$

$4 x^{3}$

$24 x^{2}$

Step 10

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

$2 x^{3}$

$4 x^{2}$

$x$

$6$

$2 x^{4}$

$8 x^{3}$

$25 x^{2}$

$6 x$

$14$

$2 x^{4}$

$12 x^{3}$

$4 x^{3}$

$25 x^{2}$

$4 x^{3}$

$24 x^{2}$

$x^{2}$

Step 11

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

$2 x^{3}$

$4 x^{2}$

$x$

$6$

$2 x^{4}$

$8 x^{3}$

$25 x^{2}$

$6 x$

$14$

$2 x^{4}$

$12 x^{3}$

$4 x^{3}$

$25 x^{2}$

$4 x^{3}$

$24 x^{2}$

$x^{2}$

$6 x$

Step 12

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

$2 x^{3}$

$4 x^{2}$

$x$

$6$

$2 x^{4}$

$8 x^{3}$

$25 x^{2}$

$6 x$

$14$

$2 x^{4}$

$12 x^{3}$

$4 x^{3}$

$25 x^{2}$

$4 x^{3}$

$24 x^{2}$

$x^{2}$

$6 x$

Step 13

Multiply the new quotient term by the divisor.

$2 x^{3}$

$4 x^{2}$

$x$

$6$

$2 x^{4}$

$8 x^{3}$

$25 x^{2}$

$6 x$

$14$

$2 x^{4}$

$12 x^{3}$

$4 x^{3}$

$25 x^{2}$

$4 x^{3}$

$24 x^{2}$

$x^{2}$

$6 x$

$x^{2}$

$6 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- x^{2} - 6 x$

$2 x^{3}$

$4 x^{2}$

$x$

$6$

$2 x^{4}$

$8 x^{3}$

$25 x^{2}$

$6 x$

$14$

$2 x^{4}$

$12 x^{3}$

$4 x^{3}$

$25 x^{2}$

$4 x^{3}$

$24 x^{2}$

$x^{2}$

$6 x$

$x^{2}$

$6 x$

Step 15

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

$2 x^{3}$

$4 x^{2}$

$x$

$6$

$2 x^{4}$

$8 x^{3}$

$25 x^{2}$

$6 x$

$14$

$2 x^{4}$

$12 x^{3}$

$4 x^{3}$

$25 x^{2}$

$4 x^{3}$

$24 x^{2}$

$x^{2}$

$6 x$

$x^{2}$

$6 x$

$0$

Step 16

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

$2 x^{3}$

$4 x^{2}$

$x$

$6$

$2 x^{4}$

$8 x^{3}$

$25 x^{2}$

$6 x$

$14$

$2 x^{4}$

$12 x^{3}$

$4 x^{3}$

$25 x^{2}$

$4 x^{3}$

$24 x^{2}$

$x^{2}$

$6 x$

$x^{2}$

$6 x$

$0$

$14$

Step 17

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

$2 x^{3} - 4 x^{2} - x + \frac{14}{x + 6}$