Algebra Examples

$(x^{4} + x^{3} - 8 x^{2} - 13 x - 7) \div (x + 2)$

Step 1

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

$x$

$2$

$x^{4}$

$x^{3}$

$8 x^{2}$

$13 x$

$7$

Step 2

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

$x^{3}$

$x$

$2$

$x^{4}$

$x^{3}$

$8 x^{2}$

$13 x$

$7$

Step 3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$2$

$x^{4}$

$x^{3}$

$8 x^{2}$

$13 x$

$7$

$x^{4}$

$2 x^{3}$

Step 4

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

				$x^{3}$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$8 x^{2}$	-	$13 x$	-	$7$
			-	$x^{4}$	-	$2 x^{3}$

Step 5

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

				$x^{3}$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$8 x^{2}$	-	$13 x$	-	$7$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$

Step 6

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

				$x^{3}$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$8 x^{2}$	-	$13 x$	-	$7$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	-	$8 x^{2}$

Step 7

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

				$x^{3}$	-	$x^{2}$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$8 x^{2}$	-	$13 x$	-	$7$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	-	$8 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$x^{2}$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$8 x^{2}$	-	$13 x$	-	$7$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	-	$8 x^{2}$
					-	$x^{3}$	-	$2 x^{2}$

Step 9

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

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$x^{3}$

$8 x^{2}$

$13 x$

$7$

$x^{4}$

$2 x^{3}$

$x^{3}$

$8 x^{2}$

$x^{3}$

$2 x^{2}$

Step 10

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

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$x^{3}$

$8 x^{2}$

$13 x$

$7$

$x^{4}$

$2 x^{3}$

$x^{3}$

$8 x^{2}$

$x^{3}$

$2 x^{2}$

$6 x^{2}$

Step 11

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

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$x^{3}$

$8 x^{2}$

$13 x$

$7$

$x^{4}$

$2 x^{3}$

$x^{3}$

$8 x^{2}$

$x^{3}$

$2 x^{2}$

$6 x^{2}$

$13 x$

Step 12

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

$x^{3}$

$x^{2}$

$6 x$

$x$

$2$

$x^{4}$

$x^{3}$

$8 x^{2}$

$13 x$

$7$

$x^{4}$

$2 x^{3}$

$x^{3}$

$8 x^{2}$

$x^{3}$

$2 x^{2}$

$6 x^{2}$

$13 x$

Step 13

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$x^{2}$	-	$6 x$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$8 x^{2}$	-	$13 x$	-	$7$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	-	$8 x^{2}$
					+	$x^{3}$	+	$2 x^{2}$
							-	$6 x^{2}$	-	$13 x$
							-	$6 x^{2}$	-	$12 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x^{2} - 12 x$

$x^{3}$

$x^{2}$

$6 x$

$x$

$2$

$x^{4}$

$x^{3}$

$8 x^{2}$

$13 x$

$7$

$x^{4}$

$2 x^{3}$

$x^{3}$

$8 x^{2}$

$x^{3}$

$2 x^{2}$

$6 x^{2}$

$13 x$

$6 x^{2}$

$12 x$

Step 15

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

$x^{3}$

$x^{2}$

$6 x$

$x$

$2$

$x^{4}$

$x^{3}$

$8 x^{2}$

$13 x$

$7$

$x^{4}$

$2 x^{3}$

$x^{3}$

$8 x^{2}$

$x^{3}$

$2 x^{2}$

$6 x^{2}$

$13 x$

$6 x^{2}$

$12 x$

$x$

Step 16

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

$x^{3}$

$x^{2}$

$6 x$

$x$

$2$

$x^{4}$

$x^{3}$

$8 x^{2}$

$13 x$

$7$

$x^{4}$

$2 x^{3}$

$x^{3}$

$8 x^{2}$

$x^{3}$

$2 x^{2}$

$6 x^{2}$

$13 x$

$6 x^{2}$

$12 x$

$x$

$7$

Step 17

Divide the highest order term in the dividend $- x$ by the highest order term in divisor $x$ .

				$x^{3}$	-	$x^{2}$	-	$6 x$	-	$1$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$8 x^{2}$	-	$13 x$	-	$7$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	-	$8 x^{2}$
					+	$x^{3}$	+	$2 x^{2}$
							-	$6 x^{2}$	-	$13 x$
							+	$6 x^{2}$	+	$12 x$
									-	$x$	-	$7$

Step 18

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$x^{2}$	-	$6 x$	-	$1$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$8 x^{2}$	-	$13 x$	-	$7$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	-	$8 x^{2}$
					+	$x^{3}$	+	$2 x^{2}$
							-	$6 x^{2}$	-	$13 x$
							+	$6 x^{2}$	+	$12 x$
									-	$x$	-	$7$
									-	$x$	-	$2$

Step 19

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

				$x^{3}$	-	$x^{2}$	-	$6 x$	-	$1$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$8 x^{2}$	-	$13 x$	-	$7$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	-	$8 x^{2}$
					+	$x^{3}$	+	$2 x^{2}$
							-	$6 x^{2}$	-	$13 x$
							+	$6 x^{2}$	+	$12 x$
									-	$x$	-	$7$
									+	$x$	+	$2$

Step 20

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

				$x^{3}$	-	$x^{2}$	-	$6 x$	-	$1$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$8 x^{2}$	-	$13 x$	-	$7$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	-	$8 x^{2}$
					+	$x^{3}$	+	$2 x^{2}$
							-	$6 x^{2}$	-	$13 x$
							+	$6 x^{2}$	+	$12 x$
									-	$x$	-	$7$
									+	$x$	+	$2$
											-	$5$

Step 21

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

$x^{3} - x^{2} - 6 x - 1 - \frac{5}{x + 2}$