Algebra Examples

$(6 x^{4} - 5 x^{3} - 18 x^{2} - 5 x + 2) \div (3 x + 2)$

Step 1

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

$3 x$

$2$

$6 x^{4}$

$5 x^{3}$

$18 x^{2}$

$5 x$

$2$

Step 2

Divide the highest order term in the dividend $6 x^{4}$ by the highest order term in divisor $3 x$ .

$2 x^{3}$

$3 x$

$2$

$6 x^{4}$

$5 x^{3}$

$18 x^{2}$

$5 x$

$2$

Step 3

Multiply the new quotient term by the divisor.

$2 x^{3}$

$3 x$

$2$

$6 x^{4}$

$5 x^{3}$

$18 x^{2}$

$5 x$

$2$

$6 x^{4}$

$4 x^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{4} + 4 x^{3}$

				$2 x^{3}$
$3 x$	+	$2$		$6 x^{4}$	-	$5 x^{3}$	-	$18 x^{2}$	-	$5 x$	+	$2$
			-	$6 x^{4}$	-	$4 x^{3}$

Step 5

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

				$2 x^{3}$
$3 x$	+	$2$		$6 x^{4}$	-	$5 x^{3}$	-	$18 x^{2}$	-	$5 x$	+	$2$
			-	$6 x^{4}$	-	$4 x^{3}$
					-	$9 x^{3}$

Step 6

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

				$2 x^{3}$
$3 x$	+	$2$		$6 x^{4}$	-	$5 x^{3}$	-	$18 x^{2}$	-	$5 x$	+	$2$
			-	$6 x^{4}$	-	$4 x^{3}$
					-	$9 x^{3}$	-	$18 x^{2}$

Step 7

Divide the highest order term in the dividend $- 9 x^{3}$ by the highest order term in divisor $3 x$ .

				$2 x^{3}$	-	$3 x^{2}$
$3 x$	+	$2$		$6 x^{4}$	-	$5 x^{3}$	-	$18 x^{2}$	-	$5 x$	+	$2$
			-	$6 x^{4}$	-	$4 x^{3}$
					-	$9 x^{3}$	-	$18 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

				$2 x^{3}$	-	$3 x^{2}$
$3 x$	+	$2$		$6 x^{4}$	-	$5 x^{3}$	-	$18 x^{2}$	-	$5 x$	+	$2$
			-	$6 x^{4}$	-	$4 x^{3}$
					-	$9 x^{3}$	-	$18 x^{2}$
					-	$9 x^{3}$	-	$6 x^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x^{3} - 6 x^{2}$

$2 x^{3}$

$3 x^{2}$

$3 x$

$2$

$6 x^{4}$

$5 x^{3}$

$18 x^{2}$

$5 x$

$2$

$6 x^{4}$

$4 x^{3}$

$9 x^{3}$

$18 x^{2}$

$9 x^{3}$

$6 x^{2}$

Step 10

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

$2 x^{3}$

$3 x^{2}$

$3 x$

$2$

$6 x^{4}$

$5 x^{3}$

$18 x^{2}$

$5 x$

$2$

$6 x^{4}$

$4 x^{3}$

$9 x^{3}$

$18 x^{2}$

$9 x^{3}$

$6 x^{2}$

$12 x^{2}$

Step 11

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

$2 x^{3}$

$3 x^{2}$

$3 x$

$2$

$6 x^{4}$

$5 x^{3}$

$18 x^{2}$

$5 x$

$2$

$6 x^{4}$

$4 x^{3}$

$9 x^{3}$

$18 x^{2}$

$9 x^{3}$

$6 x^{2}$

$12 x^{2}$

$5 x$

Step 12

Divide the highest order term in the dividend $- 12 x^{2}$ by the highest order term in divisor $3 x$ .

$2 x^{3}$

$3 x^{2}$

$4 x$

$3 x$

$2$

$6 x^{4}$

$5 x^{3}$

$18 x^{2}$

$5 x$

$2$

$6 x^{4}$

$4 x^{3}$

$9 x^{3}$

$18 x^{2}$

$9 x^{3}$

$6 x^{2}$

$12 x^{2}$

$5 x$

Step 13

Multiply the new quotient term by the divisor.

				$2 x^{3}$	-	$3 x^{2}$	-	$4 x$
$3 x$	+	$2$		$6 x^{4}$	-	$5 x^{3}$	-	$18 x^{2}$	-	$5 x$	+	$2$
			-	$6 x^{4}$	-	$4 x^{3}$
					-	$9 x^{3}$	-	$18 x^{2}$
					+	$9 x^{3}$	+	$6 x^{2}$
							-	$12 x^{2}$	-	$5 x$
							-	$12 x^{2}$	-	$8 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{2} - 8 x$

$2 x^{3}$

$3 x^{2}$

$4 x$

$3 x$

$2$

$6 x^{4}$

$5 x^{3}$

$18 x^{2}$

$5 x$

$2$

$6 x^{4}$

$4 x^{3}$

$9 x^{3}$

$18 x^{2}$

$9 x^{3}$

$6 x^{2}$

$12 x^{2}$

$5 x$

$12 x^{2}$

$8 x$

Step 15

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

$2 x^{3}$

$3 x^{2}$

$4 x$

$3 x$

$2$

$6 x^{4}$

$5 x^{3}$

$18 x^{2}$

$5 x$

$2$

$6 x^{4}$

$4 x^{3}$

$9 x^{3}$

$18 x^{2}$

$9 x^{3}$

$6 x^{2}$

$12 x^{2}$

$5 x$

$12 x^{2}$

$8 x$

$3 x$

Step 16

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

$2 x^{3}$

$3 x^{2}$

$4 x$

$3 x$

$2$

$6 x^{4}$

$5 x^{3}$

$18 x^{2}$

$5 x$

$2$

$6 x^{4}$

$4 x^{3}$

$9 x^{3}$

$18 x^{2}$

$9 x^{3}$

$6 x^{2}$

$12 x^{2}$

$5 x$

$12 x^{2}$

$8 x$

$3 x$

$2$

Step 17

Divide the highest order term in the dividend $3 x$ by the highest order term in divisor $3 x$ .

$2 x^{3}$

$3 x^{2}$

$4 x$

$1$

$3 x$

$2$

$6 x^{4}$

$5 x^{3}$

$18 x^{2}$

$5 x$

$2$

$6 x^{4}$

$4 x^{3}$

$9 x^{3}$

$18 x^{2}$

$9 x^{3}$

$6 x^{2}$

$12 x^{2}$

$5 x$

$12 x^{2}$

$8 x$

$3 x$

$2$

Step 18

Multiply the new quotient term by the divisor.

$2 x^{3}$

$3 x^{2}$

$4 x$

$1$

$3 x$

$2$

$6 x^{4}$

$5 x^{3}$

$18 x^{2}$

$5 x$

$2$

$6 x^{4}$

$4 x^{3}$

$9 x^{3}$

$18 x^{2}$

$9 x^{3}$

$6 x^{2}$

$12 x^{2}$

$5 x$

$12 x^{2}$

$8 x$

$3 x$

$2$

$3 x$

$2$

Step 19

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

$2 x^{3}$

$3 x^{2}$

$4 x$

$1$

$3 x$

$2$

$6 x^{4}$

$5 x^{3}$

$18 x^{2}$

$5 x$

$2$

$6 x^{4}$

$4 x^{3}$

$9 x^{3}$

$18 x^{2}$

$9 x^{3}$

$6 x^{2}$

$12 x^{2}$

$5 x$

$12 x^{2}$

$8 x$

$3 x$

$2$

$3 x$

$2$

Step 20

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

$2 x^{3}$

$3 x^{2}$

$4 x$

$1$

$3 x$

$2$

$6 x^{4}$

$5 x^{3}$

$18 x^{2}$

$5 x$

$2$

$6 x^{4}$

$4 x^{3}$

$9 x^{3}$

$18 x^{2}$

$9 x^{3}$

$6 x^{2}$

$12 x^{2}$

$5 x$

$12 x^{2}$

$8 x$

$3 x$

$2$

$3 x$

$2$

$0$

Step 21

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

$2 x^{3} - 3 x^{2} - 4 x + 1$