Algebra Examples

$(2 x^{4} - 8 x^{3} + 4 x^{2} - 11 x - 11) \div (x^{2} - 1)$

Step 1

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

$x^{2}$

$0 x$

$1$

$2 x^{4}$

$8 x^{3}$

$4 x^{2}$

$11 x$

$11$

Step 2

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

							$2 x^{2}$
	$x^{2}$	+	$0 x$	-	$1$		$2 x^{4}$	-	$8 x^{3}$	+	$4 x^{2}$	-	$11 x$	-	$11$

Step 3

Multiply the new quotient term by the divisor.

$2 x^{2}$

$x^{2}$

$0 x$

$1$

$2 x^{4}$

$8 x^{3}$

$4 x^{2}$

$11 x$

$11$

$2 x^{4}$

$0$

$2 x^{2}$

Step 4

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

						$2 x^{2}$
$x^{2}$	+	$0 x$	-	$1$		$2 x^{4}$	-	$8 x^{3}$	+	$4 x^{2}$	-	$11 x$	-	$11$
					-	$2 x^{4}$	-	$0$	+	$2 x^{2}$

Step 5

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

						$2 x^{2}$
$x^{2}$	+	$0 x$	-	$1$		$2 x^{4}$	-	$8 x^{3}$	+	$4 x^{2}$	-	$11 x$	-	$11$
					-	$2 x^{4}$	-	$0$	+	$2 x^{2}$
							-	$8 x^{3}$	+	$6 x^{2}$

Step 6

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

						$2 x^{2}$
$x^{2}$	+	$0 x$	-	$1$		$2 x^{4}$	-	$8 x^{3}$	+	$4 x^{2}$	-	$11 x$	-	$11$
					-	$2 x^{4}$	-	$0$	+	$2 x^{2}$
							-	$8 x^{3}$	+	$6 x^{2}$	-	$11 x$

Step 7

Divide the highest order term in the dividend $- 8 x^{3}$ by the highest order term in divisor $x^{2}$ .

						$2 x^{2}$	-	$8 x$
$x^{2}$	+	$0 x$	-	$1$		$2 x^{4}$	-	$8 x^{3}$	+	$4 x^{2}$	-	$11 x$	-	$11$
					-	$2 x^{4}$	-	$0$	+	$2 x^{2}$
							-	$8 x^{3}$	+	$6 x^{2}$	-	$11 x$

Step 8

Multiply the new quotient term by the divisor.

						$2 x^{2}$	-	$8 x$
$x^{2}$	+	$0 x$	-	$1$		$2 x^{4}$	-	$8 x^{3}$	+	$4 x^{2}$	-	$11 x$	-	$11$
					-	$2 x^{4}$	-	$0$	+	$2 x^{2}$
							-	$8 x^{3}$	+	$6 x^{2}$	-	$11 x$
							-	$8 x^{3}$	+	$0$	+	$8 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{3} + 0 + 8 x$

						$2 x^{2}$	-	$8 x$
$x^{2}$	+	$0 x$	-	$1$		$2 x^{4}$	-	$8 x^{3}$	+	$4 x^{2}$	-	$11 x$	-	$11$
					-	$2 x^{4}$	-	$0$	+	$2 x^{2}$
							-	$8 x^{3}$	+	$6 x^{2}$	-	$11 x$
							+	$8 x^{3}$	-	$0$	-	$8 x$

Step 10

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

						$2 x^{2}$	-	$8 x$
$x^{2}$	+	$0 x$	-	$1$		$2 x^{4}$	-	$8 x^{3}$	+	$4 x^{2}$	-	$11 x$	-	$11$
					-	$2 x^{4}$	-	$0$	+	$2 x^{2}$
							-	$8 x^{3}$	+	$6 x^{2}$	-	$11 x$
							+	$8 x^{3}$	-	$0$	-	$8 x$
									+	$6 x^{2}$	-	$19 x$

Step 11

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

						$2 x^{2}$	-	$8 x$
$x^{2}$	+	$0 x$	-	$1$		$2 x^{4}$	-	$8 x^{3}$	+	$4 x^{2}$	-	$11 x$	-	$11$
					-	$2 x^{4}$	-	$0$	+	$2 x^{2}$
							-	$8 x^{3}$	+	$6 x^{2}$	-	$11 x$
							+	$8 x^{3}$	-	$0$	-	$8 x$
									+	$6 x^{2}$	-	$19 x$	-	$11$

Step 12

Divide the highest order term in the dividend $6 x^{2}$ by the highest order term in divisor $x^{2}$ .

						$2 x^{2}$	-	$8 x$	+	$6$
$x^{2}$	+	$0 x$	-	$1$		$2 x^{4}$	-	$8 x^{3}$	+	$4 x^{2}$	-	$11 x$	-	$11$
					-	$2 x^{4}$	-	$0$	+	$2 x^{2}$
							-	$8 x^{3}$	+	$6 x^{2}$	-	$11 x$
							+	$8 x^{3}$	-	$0$	-	$8 x$
									+	$6 x^{2}$	-	$19 x$	-	$11$

Step 13

Multiply the new quotient term by the divisor.

						$2 x^{2}$	-	$8 x$	+	$6$
$x^{2}$	+	$0 x$	-	$1$		$2 x^{4}$	-	$8 x^{3}$	+	$4 x^{2}$	-	$11 x$	-	$11$
					-	$2 x^{4}$	-	$0$	+	$2 x^{2}$
							-	$8 x^{3}$	+	$6 x^{2}$	-	$11 x$
							+	$8 x^{3}$	-	$0$	-	$8 x$
									+	$6 x^{2}$	-	$19 x$	-	$11$
									+	$6 x^{2}$	+	$0$	-	$6$

Step 14

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

						$2 x^{2}$	-	$8 x$	+	$6$
$x^{2}$	+	$0 x$	-	$1$		$2 x^{4}$	-	$8 x^{3}$	+	$4 x^{2}$	-	$11 x$	-	$11$
					-	$2 x^{4}$	-	$0$	+	$2 x^{2}$
							-	$8 x^{3}$	+	$6 x^{2}$	-	$11 x$
							+	$8 x^{3}$	-	$0$	-	$8 x$
									+	$6 x^{2}$	-	$19 x$	-	$11$
									-	$6 x^{2}$	-	$0$	+	$6$

Step 15

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

						$2 x^{2}$	-	$8 x$	+	$6$
$x^{2}$	+	$0 x$	-	$1$		$2 x^{4}$	-	$8 x^{3}$	+	$4 x^{2}$	-	$11 x$	-	$11$
					-	$2 x^{4}$	-	$0$	+	$2 x^{2}$
							-	$8 x^{3}$	+	$6 x^{2}$	-	$11 x$
							+	$8 x^{3}$	-	$0$	-	$8 x$
									+	$6 x^{2}$	-	$19 x$	-	$11$
									-	$6 x^{2}$	-	$0$	+	$6$
											-	$19 x$	-	$5$

Step 16

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

$2 x^{2} - 8 x + 6 + \frac{- 19 x - 5}{x^{2} - 1}$