Algebra Examples

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

Step 1

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

$x$

$2$

$2 x^{3}$

$x^{2}$

$11 x$

$2$

Step 2

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

					$2 x^{2}$
	$x$	-	$2$		$2 x^{3}$	+	$x^{2}$	-	$11 x$	+	$2$

Step 3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	+	$x^{2}$	-	$11 x$	+	$2$
			+	$2 x^{3}$	-	$4 x^{2}$

Step 4

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

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	+	$x^{2}$	-	$11 x$	+	$2$
			-	$2 x^{3}$	+	$4 x^{2}$

Step 5

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

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	+	$x^{2}$	-	$11 x$	+	$2$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$5 x^{2}$

Step 6

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

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	+	$x^{2}$	-	$11 x$	+	$2$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$5 x^{2}$	-	$11 x$

Step 7

Divide the highest order term in the dividend $5 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$5 x$
$x$	-	$2$		$2 x^{3}$	+	$x^{2}$	-	$11 x$	+	$2$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$5 x^{2}$	-	$11 x$

Step 8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$5 x$
$x$	-	$2$		$2 x^{3}$	+	$x^{2}$	-	$11 x$	+	$2$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$5 x^{2}$	-	$11 x$
					+	$5 x^{2}$	-	$10 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{2} - 10 x$

				$2 x^{2}$	+	$5 x$
$x$	-	$2$		$2 x^{3}$	+	$x^{2}$	-	$11 x$	+	$2$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$5 x^{2}$	-	$11 x$
					-	$5 x^{2}$	+	$10 x$

Step 10

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

				$2 x^{2}$	+	$5 x$
$x$	-	$2$		$2 x^{3}$	+	$x^{2}$	-	$11 x$	+	$2$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$5 x^{2}$	-	$11 x$
					-	$5 x^{2}$	+	$10 x$
							-	$x$

Step 11

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

				$2 x^{2}$	+	$5 x$
$x$	-	$2$		$2 x^{3}$	+	$x^{2}$	-	$11 x$	+	$2$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$5 x^{2}$	-	$11 x$
					-	$5 x^{2}$	+	$10 x$
							-	$x$	+	$2$

Step 12

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

				$2 x^{2}$	+	$5 x$	-	$1$
$x$	-	$2$		$2 x^{3}$	+	$x^{2}$	-	$11 x$	+	$2$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$5 x^{2}$	-	$11 x$
					-	$5 x^{2}$	+	$10 x$
							-	$x$	+	$2$

Step 13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$5 x$	-	$1$
$x$	-	$2$		$2 x^{3}$	+	$x^{2}$	-	$11 x$	+	$2$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$5 x^{2}$	-	$11 x$
					-	$5 x^{2}$	+	$10 x$
							-	$x$	+	$2$
							-	$x$	+	$2$

Step 14

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

				$2 x^{2}$	+	$5 x$	-	$1$
$x$	-	$2$		$2 x^{3}$	+	$x^{2}$	-	$11 x$	+	$2$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$5 x^{2}$	-	$11 x$
					-	$5 x^{2}$	+	$10 x$
							-	$x$	+	$2$
							+	$x$	-	$2$

Step 15

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

				$2 x^{2}$	+	$5 x$	-	$1$
$x$	-	$2$		$2 x^{3}$	+	$x^{2}$	-	$11 x$	+	$2$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$5 x^{2}$	-	$11 x$
					-	$5 x^{2}$	+	$10 x$
							-	$x$	+	$2$
							+	$x$	-	$2$
										$0$

Step 16

Since the remander is $0$ , the final answer is the quotient.

$2 x^{2} + 5 x - 1$