Algebra Examples

$(2 x^{4} - 3 x^{3} - 20 x^{2} + 72 x - 63) \div (2 x - 3)$

Step 1

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

$2 x$

$3$

$2 x^{4}$

$3 x^{3}$

$20 x^{2}$

$72 x$

$63$

Step 2

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

					$x^{3}$
	$2 x$	-	$3$		$2 x^{4}$	-	$3 x^{3}$	-	$20 x^{2}$	+	$72 x$	-	$63$

Step 3

Multiply the new quotient term by the divisor.

				$x^{3}$
$2 x$	-	$3$		$2 x^{4}$	-	$3 x^{3}$	-	$20 x^{2}$	+	$72 x$	-	$63$
			+	$2 x^{4}$	-	$3 x^{3}$

Step 4

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

				$x^{3}$
$2 x$	-	$3$		$2 x^{4}$	-	$3 x^{3}$	-	$20 x^{2}$	+	$72 x$	-	$63$
			-	$2 x^{4}$	+	$3 x^{3}$

Step 5

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

				$x^{3}$
$2 x$	-	$3$		$2 x^{4}$	-	$3 x^{3}$	-	$20 x^{2}$	+	$72 x$	-	$63$
			-	$2 x^{4}$	+	$3 x^{3}$
						$0$

Step 6

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

				$x^{3}$
$2 x$	-	$3$		$2 x^{4}$	-	$3 x^{3}$	-	$20 x^{2}$	+	$72 x$	-	$63$
			-	$2 x^{4}$	+	$3 x^{3}$
						$0$	-	$20 x^{2}$	+	$72 x$

Step 7

Divide the highest order term in the dividend $- 20 x^{2}$ by the highest order term in divisor $2 x$ .

$x^{3}$

$0 x^{2}$

$10 x$

$2 x$

$3$

$2 x^{4}$

$3 x^{3}$

$20 x^{2}$

$72 x$

$63$

$2 x^{4}$

$3 x^{3}$

$0$

$20 x^{2}$

$72 x$

Step 8

Multiply the new quotient term by the divisor.

$x^{3}$

$0 x^{2}$

$10 x$

$2 x$

$3$

$2 x^{4}$

$3 x^{3}$

$20 x^{2}$

$72 x$

$63$

$2 x^{4}$

$3 x^{3}$

$0$

$20 x^{2}$

$72 x$

$20 x^{2}$

$30 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 20 x^{2} + 30 x$

$x^{3}$

$0 x^{2}$

$10 x$

$2 x$

$3$

$2 x^{4}$

$3 x^{3}$

$20 x^{2}$

$72 x$

$63$

$2 x^{4}$

$3 x^{3}$

$0$

$20 x^{2}$

$72 x$

$20 x^{2}$

$30 x$

Step 10

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

$x^{3}$

$0 x^{2}$

$10 x$

$2 x$

$3$

$2 x^{4}$

$3 x^{3}$

$20 x^{2}$

$72 x$

$63$

$2 x^{4}$

$3 x^{3}$

$0$

$20 x^{2}$

$72 x$

$20 x^{2}$

$30 x$

$42 x$

Step 11

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

$x^{3}$

$0 x^{2}$

$10 x$

$2 x$

$3$

$2 x^{4}$

$3 x^{3}$

$20 x^{2}$

$72 x$

$63$

$2 x^{4}$

$3 x^{3}$

$0$

$20 x^{2}$

$72 x$

$20 x^{2}$

$30 x$

$42 x$

$63$

Step 12

Divide the highest order term in the dividend $42 x$ by the highest order term in divisor $2 x$ .

$x^{3}$

$0 x^{2}$

$10 x$

$21$

$2 x$

$3$

$2 x^{4}$

$3 x^{3}$

$20 x^{2}$

$72 x$

$63$

$2 x^{4}$

$3 x^{3}$

$0$

$20 x^{2}$

$72 x$

$20 x^{2}$

$30 x$

$42 x$

$63$

Step 13

Multiply the new quotient term by the divisor.

				$x^{3}$	+	$0 x^{2}$	-	$10 x$	+	$21$
$2 x$	-	$3$		$2 x^{4}$	-	$3 x^{3}$	-	$20 x^{2}$	+	$72 x$	-	$63$
			-	$2 x^{4}$	+	$3 x^{3}$
						$0$	-	$20 x^{2}$	+	$72 x$
							+	$20 x^{2}$	-	$30 x$
									+	$42 x$	-	$63$
									+	$42 x$	-	$63$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $42 x - 63$

				$x^{3}$	+	$0 x^{2}$	-	$10 x$	+	$21$
$2 x$	-	$3$		$2 x^{4}$	-	$3 x^{3}$	-	$20 x^{2}$	+	$72 x$	-	$63$
			-	$2 x^{4}$	+	$3 x^{3}$
						$0$	-	$20 x^{2}$	+	$72 x$
							+	$20 x^{2}$	-	$30 x$
									+	$42 x$	-	$63$
									-	$42 x$	+	$63$

Step 15

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

				$x^{3}$	+	$0 x^{2}$	-	$10 x$	+	$21$
$2 x$	-	$3$		$2 x^{4}$	-	$3 x^{3}$	-	$20 x^{2}$	+	$72 x$	-	$63$
			-	$2 x^{4}$	+	$3 x^{3}$
						$0$	-	$20 x^{2}$	+	$72 x$
							+	$20 x^{2}$	-	$30 x$
									+	$42 x$	-	$63$
									-	$42 x$	+	$63$
												$0$

Step 16

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

$x^{3} + 0 x^{2} - 10 x + 21$