Precalculus Examples

$(h^{4} + 3 h^{3} - 5 h^{2} + 15 h - 25) \div (h^{2} + 5)$

Step 1

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

$h^{2}$

$0 h$

$5$

$h^{4}$

$3 h^{3}$

$5 h^{2}$

$15 h$

$25$

Step 2

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

$h^{2}$

$0 h$

$5$

$h^{4}$

$3 h^{3}$

$5 h^{2}$

$15 h$

$25$

Step 3

Multiply the new quotient term by the divisor.

$h^{2}$

$0 h$

$5$

$h^{4}$

$3 h^{3}$

$5 h^{2}$

$15 h$

$25$

$h^{4}$

$0$

$5 h^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $h^{4} + 0 + 5 h^{2}$

$h^{2}$

$0 h$

$5$

$h^{4}$

$3 h^{3}$

$5 h^{2}$

$15 h$

$25$

$h^{4}$

$0$

$5 h^{2}$

Step 5

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

$h^{2}$

$0 h$

$5$

$h^{4}$

$3 h^{3}$

$5 h^{2}$

$15 h$

$25$

$h^{4}$

$0$

$5 h^{2}$

$3 h^{3}$

$10 h^{2}$

Step 6

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

$h^{2}$

$0 h$

$5$

$h^{4}$

$3 h^{3}$

$5 h^{2}$

$15 h$

$25$

$h^{4}$

$0$

$5 h^{2}$

$3 h^{3}$

$10 h^{2}$

$15 h$

Step 7

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

$h^{2}$

$3 h$

$h^{2}$

$0 h$

$5$

$h^{4}$

$3 h^{3}$

$5 h^{2}$

$15 h$

$25$

$h^{4}$

$0$

$5 h^{2}$

$3 h^{3}$

$10 h^{2}$

$15 h$

Step 8

Multiply the new quotient term by the divisor.

$h^{2}$

$3 h$

$h^{2}$

$0 h$

$5$

$h^{4}$

$3 h^{3}$

$5 h^{2}$

$15 h$

$25$

$h^{4}$

$0$

$5 h^{2}$

$3 h^{3}$

$10 h^{2}$

$15 h$

$3 h^{3}$

$0$

$15 h$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $3 h^{3} + 0 + 15 h$

$h^{2}$

$3 h$

$h^{2}$

$0 h$

$5$

$h^{4}$

$3 h^{3}$

$5 h^{2}$

$15 h$

$25$

$h^{4}$

$0$

$5 h^{2}$

$3 h^{3}$

$10 h^{2}$

$15 h$

$3 h^{3}$

$0$

$15 h$

Step 10

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

$h^{2}$

$3 h$

$h^{2}$

$0 h$

$5$

$h^{4}$

$3 h^{3}$

$5 h^{2}$

$15 h$

$25$

$h^{4}$

$0$

$5 h^{2}$

$3 h^{3}$

$10 h^{2}$

$15 h$

$3 h^{3}$

$0$

$15 h$

$10 h^{2}$

$0$

Step 11

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

						$h^{2}$	+	$3 h$
$h^{2}$	+	$0 h$	+	$5$		$h^{4}$	+	$3 h^{3}$	-	$5 h^{2}$	+	$15 h$	-	$25$
					-	$h^{4}$	-	$0$	-	$5 h^{2}$
							+	$3 h^{3}$	-	$10 h^{2}$	+	$15 h$
							-	$3 h^{3}$	-	$0$	-	$15 h$
									-	$10 h^{2}$	+	$0$	-	$25$

Step 12

Divide the highest order term in the dividend $- 10 h^{2}$ by the highest order term in divisor $h^{2}$ .

						$h^{2}$	+	$3 h$	-	$10$
$h^{2}$	+	$0 h$	+	$5$		$h^{4}$	+	$3 h^{3}$	-	$5 h^{2}$	+	$15 h$	-	$25$
					-	$h^{4}$	-	$0$	-	$5 h^{2}$
							+	$3 h^{3}$	-	$10 h^{2}$	+	$15 h$
							-	$3 h^{3}$	-	$0$	-	$15 h$
									-	$10 h^{2}$	+	$0$	-	$25$

Step 13

Multiply the new quotient term by the divisor.

						$h^{2}$	+	$3 h$	-	$10$
$h^{2}$	+	$0 h$	+	$5$		$h^{4}$	+	$3 h^{3}$	-	$5 h^{2}$	+	$15 h$	-	$25$
					-	$h^{4}$	-	$0$	-	$5 h^{2}$
							+	$3 h^{3}$	-	$10 h^{2}$	+	$15 h$
							-	$3 h^{3}$	-	$0$	-	$15 h$
									-	$10 h^{2}$	+	$0$	-	$25$
									-	$10 h^{2}$	+	$0$	-	$50$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 h^{2} + 0 - 50$

						$h^{2}$	+	$3 h$	-	$10$
$h^{2}$	+	$0 h$	+	$5$		$h^{4}$	+	$3 h^{3}$	-	$5 h^{2}$	+	$15 h$	-	$25$
					-	$h^{4}$	-	$0$	-	$5 h^{2}$
							+	$3 h^{3}$	-	$10 h^{2}$	+	$15 h$
							-	$3 h^{3}$	-	$0$	-	$15 h$
									-	$10 h^{2}$	+	$0$	-	$25$
									+	$10 h^{2}$	-	$0$	+	$50$

Step 15

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

						$h^{2}$	+	$3 h$	-	$10$
$h^{2}$	+	$0 h$	+	$5$		$h^{4}$	+	$3 h^{3}$	-	$5 h^{2}$	+	$15 h$	-	$25$
					-	$h^{4}$	-	$0$	-	$5 h^{2}$
							+	$3 h^{3}$	-	$10 h^{2}$	+	$15 h$
							-	$3 h^{3}$	-	$0$	-	$15 h$
									-	$10 h^{2}$	+	$0$	-	$25$
									+	$10 h^{2}$	-	$0$	+	$50$
													+	$25$

Step 16

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

$h^{2} + 3 h - 10 + \frac{25}{h^{2} + 5}$