Algebra Examples

$\frac{2 h^{4} - h^{3} + h^{2} + h - 3}{h^{2} - 1}$

Step 1

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

$h^{2}$

$0 h$

$1$

$2 h^{4}$

$h^{3}$

$h^{2}$

$h$

$3$

Step 2

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

$2 h^{2}$

$h^{2}$

$0 h$

$1$

$2 h^{4}$

$h^{3}$

$h^{2}$

$h$

$3$

Step 3

Multiply the new quotient term by the divisor.

$2 h^{2}$

$h^{2}$

$0 h$

$1$

$2 h^{4}$

$h^{3}$

$h^{2}$

$h$

$3$

$2 h^{4}$

$0$

$2 h^{2}$

Step 4

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

$2 h^{2}$

$h^{2}$

$0 h$

$1$

$2 h^{4}$

$h^{3}$

$h^{2}$

$h$

$3$

$2 h^{4}$

$0$

$2 h^{2}$

Step 5

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

$2 h^{2}$

$h^{2}$

$0 h$

$1$

$2 h^{4}$

$h^{3}$

$h^{2}$

$h$

$3$

$2 h^{4}$

$0$

$2 h^{2}$

$h^{3}$

$3 h^{2}$

Step 6

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

$2 h^{2}$

$h^{2}$

$0 h$

$1$

$2 h^{4}$

$h^{3}$

$h^{2}$

$h$

$3$

$2 h^{4}$

$0$

$2 h^{2}$

$h^{3}$

$3 h^{2}$

$h$

Step 7

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

$2 h^{2}$

$h$

$h^{2}$

$0 h$

$1$

$2 h^{4}$

$h^{3}$

$h^{2}$

$h$

$3$

$2 h^{4}$

$0$

$2 h^{2}$

$h^{3}$

$3 h^{2}$

$h$

Step 8

Multiply the new quotient term by the divisor.

$2 h^{2}$

$h$

$h^{2}$

$0 h$

$1$

$2 h^{4}$

$h^{3}$

$h^{2}$

$h$

$3$

$2 h^{4}$

$0$

$2 h^{2}$

$h^{3}$

$3 h^{2}$

$h$

$h^{3}$

$0$

$h$

Step 9

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

$2 h^{2}$

$h$

$h^{2}$

$0 h$

$1$

$2 h^{4}$

$h^{3}$

$h^{2}$

$h$

$3$

$2 h^{4}$

$0$

$2 h^{2}$

$h^{3}$

$3 h^{2}$

$h$

$h^{3}$

$0$

$h$

Step 10

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

$2 h^{2}$

$h$

$h^{2}$

$0 h$

$1$

$2 h^{4}$

$h^{3}$

$h^{2}$

$h$

$3$

$2 h^{4}$

$0$

$2 h^{2}$

$h^{3}$

$3 h^{2}$

$h$

$h^{3}$

$0$

$h$

$3 h^{2}$

$0$

Step 11

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

$2 h^{2}$

$h$

$h^{2}$

$0 h$

$1$

$2 h^{4}$

$h^{3}$

$h^{2}$

$h$

$3$

$2 h^{4}$

$0$

$2 h^{2}$

$h^{3}$

$3 h^{2}$

$h$

$h^{3}$

$0$

$h$

$3 h^{2}$

$0$

$3$

Step 12

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

$2 h^{2}$

$h$

$3$

$h^{2}$

$0 h$

$1$

$2 h^{4}$

$h^{3}$

$h^{2}$

$h$

$3$

$2 h^{4}$

$0$

$2 h^{2}$

$h^{3}$

$3 h^{2}$

$h$

$h^{3}$

$0$

$h$

$3 h^{2}$

$0$

$3$

Step 13

Multiply the new quotient term by the divisor.

						$2 h^{2}$	-	$h$	+	$3$
$h^{2}$	+	$0 h$	-	$1$		$2 h^{4}$	-	$h^{3}$	+	$h^{2}$	+	$h$	-	$3$
					-	$2 h^{4}$	-	$0$	+	$2 h^{2}$
							-	$h^{3}$	+	$3 h^{2}$	+	$h$
							+	$h^{3}$	-	$0$	-	$h$
									+	$3 h^{2}$	+	$0$	-	$3$
									+	$3 h^{2}$	+	$0$	-	$3$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $3 h^{2} + 0 - 3$

						$2 h^{2}$	-	$h$	+	$3$
$h^{2}$	+	$0 h$	-	$1$		$2 h^{4}$	-	$h^{3}$	+	$h^{2}$	+	$h$	-	$3$
					-	$2 h^{4}$	-	$0$	+	$2 h^{2}$
							-	$h^{3}$	+	$3 h^{2}$	+	$h$
							+	$h^{3}$	-	$0$	-	$h$
									+	$3 h^{2}$	+	$0$	-	$3$
									-	$3 h^{2}$	-	$0$	+	$3$

Step 15

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

						$2 h^{2}$	-	$h$	+	$3$
$h^{2}$	+	$0 h$	-	$1$		$2 h^{4}$	-	$h^{3}$	+	$h^{2}$	+	$h$	-	$3$
					-	$2 h^{4}$	-	$0$	+	$2 h^{2}$
							-	$h^{3}$	+	$3 h^{2}$	+	$h$
							+	$h^{3}$	-	$0$	-	$h$
									+	$3 h^{2}$	+	$0$	-	$3$
									-	$3 h^{2}$	-	$0$	+	$3$
														$0$

Step 16

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

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