Algebra Examples

$\frac{x^{4} - 2 x^{3} - 8 x^{2} + 22 x + 6}{x^{2} - 6}$

Step 1

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

$x^{2}$

$0 x$

$6$

$x^{4}$

$2 x^{3}$

$8 x^{2}$

$22 x$

$6$

Step 2

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

$x^{2}$

$0 x$

$6$

$x^{4}$

$2 x^{3}$

$8 x^{2}$

$22 x$

$6$

Step 3

Multiply the new quotient term by the divisor.

$x^{2}$

$0 x$

$6$

$x^{4}$

$2 x^{3}$

$8 x^{2}$

$22 x$

$6$

$x^{4}$

$0$

$6 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 0 - 6 x^{2}$

$x^{2}$

$0 x$

$6$

$x^{4}$

$2 x^{3}$

$8 x^{2}$

$22 x$

$6$

$x^{4}$

$0$

$6 x^{2}$

Step 5

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

						$x^{2}$
$x^{2}$	+	$0 x$	-	$6$		$x^{4}$	-	$2 x^{3}$	-	$8 x^{2}$	+	$22 x$	+	$6$
					-	$x^{4}$	-	$0$	+	$6 x^{2}$
							-	$2 x^{3}$	-	$2 x^{2}$

Step 6

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

$x^{2}$

$0 x$

$6$

$x^{4}$

$2 x^{3}$

$8 x^{2}$

$22 x$

$6$

$x^{4}$

$0$

$6 x^{2}$

$2 x^{3}$

$2 x^{2}$

$22 x$

Step 7

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

$x^{2}$

$2 x$

$x^{2}$

$0 x$

$6$

$x^{4}$

$2 x^{3}$

$8 x^{2}$

$22 x$

$6$

$x^{4}$

$0$

$6 x^{2}$

$2 x^{3}$

$2 x^{2}$

$22 x$

Step 8

Multiply the new quotient term by the divisor.

$x^{2}$

$2 x$

$x^{2}$

$0 x$

$6$

$x^{4}$

$2 x^{3}$

$8 x^{2}$

$22 x$

$6$

$x^{4}$

$0$

$6 x^{2}$

$2 x^{3}$

$2 x^{2}$

$22 x$

$2 x^{3}$

$0$

$12 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{3} + 0 + 12 x$

						$x^{2}$	-	$2 x$
$x^{2}$	+	$0 x$	-	$6$		$x^{4}$	-	$2 x^{3}$	-	$8 x^{2}$	+	$22 x$	+	$6$
					-	$x^{4}$	-	$0$	+	$6 x^{2}$
							-	$2 x^{3}$	-	$2 x^{2}$	+	$22 x$
							+	$2 x^{3}$	-	$0$	-	$12 x$

Step 10

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

$x^{2}$

$2 x$

$x^{2}$

$0 x$

$6$

$x^{4}$

$2 x^{3}$

$8 x^{2}$

$22 x$

$6$

$x^{4}$

$0$

$6 x^{2}$

$2 x^{3}$

$2 x^{2}$

$22 x$

$2 x^{3}$

$0$

$12 x$

$2 x^{2}$

$10 x$

Step 11

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

$x^{2}$

$2 x$

$x^{2}$

$0 x$

$6$

$x^{4}$

$2 x^{3}$

$8 x^{2}$

$22 x$

$6$

$x^{4}$

$0$

$6 x^{2}$

$2 x^{3}$

$2 x^{2}$

$22 x$

$2 x^{3}$

$0$

$12 x$

$2 x^{2}$

$10 x$

$6$

Step 12

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

						$x^{2}$	-	$2 x$	-	$2$
$x^{2}$	+	$0 x$	-	$6$		$x^{4}$	-	$2 x^{3}$	-	$8 x^{2}$	+	$22 x$	+	$6$
					-	$x^{4}$	-	$0$	+	$6 x^{2}$
							-	$2 x^{3}$	-	$2 x^{2}$	+	$22 x$
							+	$2 x^{3}$	-	$0$	-	$12 x$
									-	$2 x^{2}$	+	$10 x$	+	$6$

Step 13

Multiply the new quotient term by the divisor.

						$x^{2}$	-	$2 x$	-	$2$
$x^{2}$	+	$0 x$	-	$6$		$x^{4}$	-	$2 x^{3}$	-	$8 x^{2}$	+	$22 x$	+	$6$
					-	$x^{4}$	-	$0$	+	$6 x^{2}$
							-	$2 x^{3}$	-	$2 x^{2}$	+	$22 x$
							+	$2 x^{3}$	-	$0$	-	$12 x$
									-	$2 x^{2}$	+	$10 x$	+	$6$
									-	$2 x^{2}$	+	$0$	+	$12$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{2} + 0 + 12$

						$x^{2}$	-	$2 x$	-	$2$
$x^{2}$	+	$0 x$	-	$6$		$x^{4}$	-	$2 x^{3}$	-	$8 x^{2}$	+	$22 x$	+	$6$
					-	$x^{4}$	-	$0$	+	$6 x^{2}$
							-	$2 x^{3}$	-	$2 x^{2}$	+	$22 x$
							+	$2 x^{3}$	-	$0$	-	$12 x$
									-	$2 x^{2}$	+	$10 x$	+	$6$
									+	$2 x^{2}$	-	$0$	-	$12$

Step 15

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

						$x^{2}$	-	$2 x$	-	$2$
$x^{2}$	+	$0 x$	-	$6$		$x^{4}$	-	$2 x^{3}$	-	$8 x^{2}$	+	$22 x$	+	$6$
					-	$x^{4}$	-	$0$	+	$6 x^{2}$
							-	$2 x^{3}$	-	$2 x^{2}$	+	$22 x$
							+	$2 x^{3}$	-	$0$	-	$12 x$
									-	$2 x^{2}$	+	$10 x$	+	$6$
									+	$2 x^{2}$	-	$0$	-	$12$
											+	$10 x$	-	$6$

Step 16

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

$x^{2} - 2 x - 2 + \frac{10 x - 6}{x^{2} - 6}$