Algebra Examples

$(27 x^{5} + 9 x^{4} - 18 x^{3}) \div (9 x^{2})$

Step 1

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

$9 x^{2}$

$0 x$

$0$

$27 x^{5}$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 2

Divide the highest order term in the dividend $27 x^{5}$ by the highest order term in divisor $9 x^{2}$ .

$3 x^{3}$

$9 x^{2}$

$0 x$

$0$

$27 x^{5}$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$3 x^{3}$

$9 x^{2}$

$0 x$

$0$

$27 x^{5}$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$0 x$

$0$

$27 x^{5}$

$0$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $27 x^{5} + 0 + 0$

$3 x^{3}$

$9 x^{2}$

$0 x$

$0$

$27 x^{5}$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$0 x$

$0$

$27 x^{5}$

$0$

Step 5

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

$3 x^{3}$

$9 x^{2}$

$0 x$

$0$

$27 x^{5}$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$0 x$

$0$

$27 x^{5}$

$0$

$9 x^{4}$

$18 x^{3}$

Step 6

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

$3 x^{3}$

$9 x^{2}$

$0 x$

$0$

$27 x^{5}$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$0 x$

$0$

$27 x^{5}$

$0$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

Step 7

Divide the highest order term in the dividend $9 x^{4}$ by the highest order term in divisor $9 x^{2}$ .

$3 x^{3}$

$x^{2}$

$9 x^{2}$

$0 x$

$0$

$27 x^{5}$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$0 x$

$0$

$27 x^{5}$

$0$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x^{2}$

$9 x^{2}$

$0 x$

$0$

$27 x^{5}$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$0 x$

$0$

$27 x^{5}$

$0$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$9 x^{4}$

$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{4} + 0 + 0$

$3 x^{3}$

$x^{2}$

$9 x^{2}$

$0 x$

$0$

$27 x^{5}$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$0 x$

$0$

$27 x^{5}$

$0$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$9 x^{4}$

$0$

Step 10

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

$3 x^{3}$

$x^{2}$

$9 x^{2}$

$0 x$

$0$

$27 x^{5}$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$0 x$

$0$

$27 x^{5}$

$0$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$9 x^{4}$

$0$

$18 x^{3}$

$0$

Step 11

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

$3 x^{3}$

$x^{2}$

$9 x^{2}$

$0 x$

$0$

$27 x^{5}$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$0 x$

$0$

$27 x^{5}$

$0$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$9 x^{4}$

$0$

$18 x^{3}$

$0$

$0 x$

$0$

Step 12

Divide the highest order term in the dividend $- 18 x^{3}$ by the highest order term in divisor $9 x^{2}$ .

$3 x^{3}$

$x^{2}$

$2 x$

$9 x^{2}$

$0 x$

$0$

$27 x^{5}$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$0 x$

$0$

$27 x^{5}$

$0$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$9 x^{4}$

$0$

$18 x^{3}$

$0$

$0 x$

$0$

Step 13

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x^{2}$

$2 x$

$9 x^{2}$

$0 x$

$0$

$27 x^{5}$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$0 x$

$0$

$27 x^{5}$

$0$

$9 x^{4}$

$18 x^{3}$

$0 x^{2}$

$9 x^{4}$

$0$

$18 x^{3}$

$0$

$0 x$

$0$

$18 x^{3}$

$0$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 18 x^{3} + 0 + 0$

						$3 x^{3}$	+	$x^{2}$	-	$2 x$
$9 x^{2}$	+	$0 x$	+	$0$		$27 x^{5}$	+	$9 x^{4}$	-	$18 x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$0$
					-	$27 x^{5}$	-	$0$	-	$0$
							+	$9 x^{4}$	-	$18 x^{3}$	+	$0 x^{2}$
							-	$9 x^{4}$	-	$0$	-	$0$
									-	$18 x^{3}$	+	$0$	+	$0 x$	+	$0$
									+	$18 x^{3}$	-	$0$	-	$0$

Step 15

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

						$3 x^{3}$	+	$x^{2}$	-	$2 x$
$9 x^{2}$	+	$0 x$	+	$0$		$27 x^{5}$	+	$9 x^{4}$	-	$18 x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$0$
					-	$27 x^{5}$	-	$0$	-	$0$
							+	$9 x^{4}$	-	$18 x^{3}$	+	$0 x^{2}$
							-	$9 x^{4}$	-	$0$	-	$0$
									-	$18 x^{3}$	+	$0$	+	$0 x$	+	$0$
									+	$18 x^{3}$	-	$0$	-	$0$
														$0$

Step 16

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

						$3 x^{3}$	+	$x^{2}$	-	$2 x$
$9 x^{2}$	+	$0 x$	+	$0$		$27 x^{5}$	+	$9 x^{4}$	-	$18 x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$0$
					-	$27 x^{5}$	-	$0$	-	$0$
							+	$9 x^{4}$	-	$18 x^{3}$	+	$0 x^{2}$
							-	$9 x^{4}$	-	$0$	-	$0$
									-	$18 x^{3}$	+	$0$	+	$0 x$	+	$0$
									+	$18 x^{3}$	-	$0$	-	$0$
														$0$	+	$0$

Step 17

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

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