Pre-Algebra Examples

$\frac{18 x^{4} + 27 x^{3} - 9 x^{2}}{3 x^{2} + 1}$

Step 1

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

$3 x^{2}$

$0 x$

$1$

$18 x^{4}$

$27 x^{3}$

$9 x^{2}$

$0 x$

$0$

Step 2

Divide the highest order term in the dividend $18 x^{4}$ by the highest order term in divisor $3 x^{2}$ .

$6 x^{2}$

$3 x^{2}$

$0 x$

$1$

$18 x^{4}$

$27 x^{3}$

$9 x^{2}$

$0 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$6 x^{2}$

$3 x^{2}$

$0 x$

$1$

$18 x^{4}$

$27 x^{3}$

$9 x^{2}$

$0 x$

$0$

$18 x^{4}$

$0$

$6 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $18 x^{4} + 0 + 6 x^{2}$

$6 x^{2}$

$3 x^{2}$

$0 x$

$1$

$18 x^{4}$

$27 x^{3}$

$9 x^{2}$

$0 x$

$0$

$18 x^{4}$

$0$

$6 x^{2}$

Step 5

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

$6 x^{2}$

$3 x^{2}$

$0 x$

$1$

$18 x^{4}$

$27 x^{3}$

$9 x^{2}$

$0 x$

$0$

$18 x^{4}$

$0$

$6 x^{2}$

$27 x^{3}$

$15 x^{2}$

Step 6

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

$6 x^{2}$

$3 x^{2}$

$0 x$

$1$

$18 x^{4}$

$27 x^{3}$

$9 x^{2}$

$0 x$

$0$

$18 x^{4}$

$0$

$6 x^{2}$

$27 x^{3}$

$15 x^{2}$

$0 x$

Step 7

Divide the highest order term in the dividend $27 x^{3}$ by the highest order term in divisor $3 x^{2}$ .

$6 x^{2}$

$9 x$

$3 x^{2}$

$0 x$

$1$

$18 x^{4}$

$27 x^{3}$

$9 x^{2}$

$0 x$

$0$

$18 x^{4}$

$0$

$6 x^{2}$

$27 x^{3}$

$15 x^{2}$

$0 x$

Step 8

Multiply the new quotient term by the divisor.

$6 x^{2}$

$9 x$

$3 x^{2}$

$0 x$

$1$

$18 x^{4}$

$27 x^{3}$

$9 x^{2}$

$0 x$

$0$

$18 x^{4}$

$0$

$6 x^{2}$

$27 x^{3}$

$15 x^{2}$

$0 x$

$27 x^{3}$

$0$

$9 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $27 x^{3} + 0 + 9 x$

$6 x^{2}$

$9 x$

$3 x^{2}$

$0 x$

$1$

$18 x^{4}$

$27 x^{3}$

$9 x^{2}$

$0 x$

$0$

$18 x^{4}$

$0$

$6 x^{2}$

$27 x^{3}$

$15 x^{2}$

$0 x$

$27 x^{3}$

$0$

$9 x$

Step 10

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

$6 x^{2}$

$9 x$

$3 x^{2}$

$0 x$

$1$

$18 x^{4}$

$27 x^{3}$

$9 x^{2}$

$0 x$

$0$

$18 x^{4}$

$0$

$6 x^{2}$

$27 x^{3}$

$15 x^{2}$

$0 x$

$27 x^{3}$

$0$

$9 x$

$15 x^{2}$

$9 x$

Step 11

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

$6 x^{2}$

$9 x$

$3 x^{2}$

$0 x$

$1$

$18 x^{4}$

$27 x^{3}$

$9 x^{2}$

$0 x$

$0$

$18 x^{4}$

$0$

$6 x^{2}$

$27 x^{3}$

$15 x^{2}$

$0 x$

$27 x^{3}$

$0$

$9 x$

$15 x^{2}$

$9 x$

$0$

Step 12

Divide the highest order term in the dividend $- 15 x^{2}$ by the highest order term in divisor $3 x^{2}$ .

$6 x^{2}$

$9 x$

$5$

$3 x^{2}$

$0 x$

$1$

$18 x^{4}$

$27 x^{3}$

$9 x^{2}$

$0 x$

$0$

$18 x^{4}$

$0$

$6 x^{2}$

$27 x^{3}$

$15 x^{2}$

$0 x$

$27 x^{3}$

$0$

$9 x$

$15 x^{2}$

$9 x$

$0$

Step 13

Multiply the new quotient term by the divisor.

						$6 x^{2}$	+	$9 x$	-	$5$
$3 x^{2}$	+	$0 x$	+	$1$		$18 x^{4}$	+	$27 x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$0$
					-	$18 x^{4}$	-	$0$	-	$6 x^{2}$
							+	$27 x^{3}$	-	$15 x^{2}$	+	$0 x$
							-	$27 x^{3}$	-	$0$	-	$9 x$
									-	$15 x^{2}$	-	$9 x$	+	$0$
									-	$15 x^{2}$	+	$0$	-	$5$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x^{2} + 0 - 5$

						$6 x^{2}$	+	$9 x$	-	$5$
$3 x^{2}$	+	$0 x$	+	$1$		$18 x^{4}$	+	$27 x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$0$
					-	$18 x^{4}$	-	$0$	-	$6 x^{2}$
							+	$27 x^{3}$	-	$15 x^{2}$	+	$0 x$
							-	$27 x^{3}$	-	$0$	-	$9 x$
									-	$15 x^{2}$	-	$9 x$	+	$0$
									+	$15 x^{2}$	-	$0$	+	$5$

Step 15

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

						$6 x^{2}$	+	$9 x$	-	$5$
$3 x^{2}$	+	$0 x$	+	$1$		$18 x^{4}$	+	$27 x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$0$
					-	$18 x^{4}$	-	$0$	-	$6 x^{2}$
							+	$27 x^{3}$	-	$15 x^{2}$	+	$0 x$
							-	$27 x^{3}$	-	$0$	-	$9 x$
									-	$15 x^{2}$	-	$9 x$	+	$0$
									+	$15 x^{2}$	-	$0$	+	$5$
											-	$9 x$	+	$5$

Step 16

The final answer is the quotient plus the remainder over the divisor.

$6 x^{2} + 9 x - 5 + \frac{- 9 x + 5}{3 x^{2} + 1}$