Algebra Examples

$2 x^{4} - 3 x^{3} - 24 x^{2} + 76 x - 60$ by $2 x - 3$

Step 1

Write the problem as a mathematical expression.

$\frac{2 x^{4} - 3 x^{3} - 24 x^{2} + 76 x - 60}{2 x - 3}$

Step 2

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

$2 x$

$3$

$2 x^{4}$

$3 x^{3}$

$24 x^{2}$

$76 x$

$60$

Step 3

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

					$x^{3}$
	$2 x$	-	$3$		$2 x^{4}$	-	$3 x^{3}$	-	$24 x^{2}$	+	$76 x$	-	$60$

Step 4

Multiply the new quotient term by the divisor.

				$x^{3}$
$2 x$	-	$3$		$2 x^{4}$	-	$3 x^{3}$	-	$24 x^{2}$	+	$76 x$	-	$60$
			+	$2 x^{4}$	-	$3 x^{3}$

Step 5

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

				$x^{3}$
$2 x$	-	$3$		$2 x^{4}$	-	$3 x^{3}$	-	$24 x^{2}$	+	$76 x$	-	$60$
			-	$2 x^{4}$	+	$3 x^{3}$

Step 6

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

				$x^{3}$
$2 x$	-	$3$		$2 x^{4}$	-	$3 x^{3}$	-	$24 x^{2}$	+	$76 x$	-	$60$
			-	$2 x^{4}$	+	$3 x^{3}$
						$0$

Step 7

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

				$x^{3}$
$2 x$	-	$3$		$2 x^{4}$	-	$3 x^{3}$	-	$24 x^{2}$	+	$76 x$	-	$60$
			-	$2 x^{4}$	+	$3 x^{3}$
						$0$	-	$24 x^{2}$	+	$76 x$

Step 8

Divide the highest order term in the dividend $- 24 x^{2}$ by the highest order term in divisor $2 x$ .

$x^{3}$

$0 x^{2}$

$12 x$

$2 x$

$3$

$2 x^{4}$

$3 x^{3}$

$24 x^{2}$

$76 x$

$60$

$2 x^{4}$

$3 x^{3}$

$0$

$24 x^{2}$

$76 x$

Step 9

Multiply the new quotient term by the divisor.

$x^{3}$

$0 x^{2}$

$12 x$

$2 x$

$3$

$2 x^{4}$

$3 x^{3}$

$24 x^{2}$

$76 x$

$60$

$2 x^{4}$

$3 x^{3}$

$0$

$24 x^{2}$

$76 x$

$24 x^{2}$

$36 x$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $- 24 x^{2} + 36 x$

$x^{3}$

$0 x^{2}$

$12 x$

$2 x$

$3$

$2 x^{4}$

$3 x^{3}$

$24 x^{2}$

$76 x$

$60$

$2 x^{4}$

$3 x^{3}$

$0$

$24 x^{2}$

$76 x$

$24 x^{2}$

$36 x$

Step 11

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

$x^{3}$

$0 x^{2}$

$12 x$

$2 x$

$3$

$2 x^{4}$

$3 x^{3}$

$24 x^{2}$

$76 x$

$60$

$2 x^{4}$

$3 x^{3}$

$0$

$24 x^{2}$

$76 x$

$24 x^{2}$

$36 x$

$40 x$

Step 12

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

$x^{3}$

$0 x^{2}$

$12 x$

$2 x$

$3$

$2 x^{4}$

$3 x^{3}$

$24 x^{2}$

$76 x$

$60$

$2 x^{4}$

$3 x^{3}$

$0$

$24 x^{2}$

$76 x$

$24 x^{2}$

$36 x$

$40 x$

$60$

Step 13

Divide the highest order term in the dividend $40 x$ by the highest order term in divisor $2 x$ .

$x^{3}$

$0 x^{2}$

$12 x$

$20$

$2 x$

$3$

$2 x^{4}$

$3 x^{3}$

$24 x^{2}$

$76 x$

$60$

$2 x^{4}$

$3 x^{3}$

$0$

$24 x^{2}$

$76 x$

$24 x^{2}$

$36 x$

$40 x$

$60$

Step 14

Multiply the new quotient term by the divisor.

				$x^{3}$	+	$0 x^{2}$	-	$12 x$	+	$20$
$2 x$	-	$3$		$2 x^{4}$	-	$3 x^{3}$	-	$24 x^{2}$	+	$76 x$	-	$60$
			-	$2 x^{4}$	+	$3 x^{3}$
						$0$	-	$24 x^{2}$	+	$76 x$
							+	$24 x^{2}$	-	$36 x$
									+	$40 x$	-	$60$
									+	$40 x$	-	$60$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $40 x - 60$

				$x^{3}$	+	$0 x^{2}$	-	$12 x$	+	$20$
$2 x$	-	$3$		$2 x^{4}$	-	$3 x^{3}$	-	$24 x^{2}$	+	$76 x$	-	$60$
			-	$2 x^{4}$	+	$3 x^{3}$
						$0$	-	$24 x^{2}$	+	$76 x$
							+	$24 x^{2}$	-	$36 x$
									+	$40 x$	-	$60$
									-	$40 x$	+	$60$

Step 16

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

				$x^{3}$	+	$0 x^{2}$	-	$12 x$	+	$20$
$2 x$	-	$3$		$2 x^{4}$	-	$3 x^{3}$	-	$24 x^{2}$	+	$76 x$	-	$60$
			-	$2 x^{4}$	+	$3 x^{3}$
						$0$	-	$24 x^{2}$	+	$76 x$
							+	$24 x^{2}$	-	$36 x$
									+	$40 x$	-	$60$
									-	$40 x$	+	$60$
												$0$

Step 17

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

$x^{3} + 0 x^{2} - 12 x + 20$