Algebra Examples

$(4 a^{2} h^{2} - 8 a^{3} h + 3 a^{4}) \div (2 a^{2})$

Step 1

Expand $4 a^{2} h^{2} - 8 a^{3} h + 3 a^{4}$ .

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Step 1.1

Move $- 8 a^{3} h$ .

$\frac{4 a^{2} h^{2} + 3 a^{4} - 8 a^{3} h}{2 a^{2}}$

Step 1.2

Reorder $4 a^{2} h^{2}$ and $3 a^{4}$ .

$\frac{3 a^{4} + 4 a^{2} h^{2} - 8 a^{3} h}{2 a^{2}}$

Step 2

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

$2 a^{2}$

$0 a$

$0$

$3 a^{4}$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$0$

Step 3

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

$\frac{3 a^{2}}{2}$

$2 a^{2}$

$0 a$

$0$

$3 a^{4}$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$0$

Step 4

Multiply the new quotient term by the divisor.

$\frac{3 a^{2}}{2}$

$2 a^{2}$

$0 a$

$0$

$3 a^{4}$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$0$

$3 a^{4}$

$0$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $3 a^{4} + 0 + 0$

$\frac{3 a^{2}}{2}$

$2 a^{2}$

$0 a$

$0$

$3 a^{4}$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$0$

$3 a^{4}$

$0$

Step 6

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

$\frac{3 a^{2}}{2}$

$2 a^{2}$

$0 a$

$0$

$3 a^{4}$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$0$

$3 a^{4}$

$0$

$8 a^{3} h$

$4 a^{2} h^{2}$

Step 7

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

$\frac{3 a^{2}}{2}$

$2 a^{2}$

$0 a$

$0$

$3 a^{4}$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$0$

$3 a^{4}$

$0$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

Step 8

Divide the highest order term in the dividend $- 8 a^{3} h$ by the highest order term in divisor $2 a^{2}$ .

$\frac{3 a^{2}}{2}$

$4 a h$

$2 a^{2}$

$0 a$

$0$

$3 a^{4}$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$0$

$3 a^{4}$

$0$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

Step 9

Multiply the new quotient term by the divisor.

$\frac{3 a^{2}}{2}$

$4 a h$

$2 a^{2}$

$0 a$

$0$

$3 a^{4}$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$0$

$3 a^{4}$

$0$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$8 a^{3} h$

$0$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 a^{3} h + 0 + 0$

						$\frac{3 a^{2}}{2}$	-	$4 a h$
$2 a^{2}$	+	$0 a$	+	$0$		$3 a^{4}$	-	$8 a^{3} h$	+	$4 a^{2} h^{2}$	+	$0 a$	+	$0$
					-	$3 a^{4}$	-	$0$	-	$0$
							-	$8 a^{3} h$	+	$4 a^{2} h^{2}$	+	$0 a$
							+	$8 a^{3} h$	-	$0$	-	$0$

Step 11

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

$\frac{3 a^{2}}{2}$

$4 a h$

$2 a^{2}$

$0 a$

$0$

$3 a^{4}$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$0$

$3 a^{4}$

$0$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$8 a^{3} h$

$0$

$4 a^{2} h^{2}$

$0$

Step 12

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

$\frac{3 a^{2}}{2}$

$4 a h$

$2 a^{2}$

$0 a$

$0$

$3 a^{4}$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$0$

$3 a^{4}$

$0$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$8 a^{3} h$

$0$

$4 a^{2} h^{2}$

$0$

Step 13

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

$\frac{3 a^{2}}{2}$

$4 a h$

$2 h^{2}$

$2 a^{2}$

$0 a$

$0$

$3 a^{4}$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$0$

$3 a^{4}$

$0$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$8 a^{3} h$

$0$

$4 a^{2} h^{2}$

$0$

Step 14

Multiply the new quotient term by the divisor.

$\frac{3 a^{2}}{2}$

$4 a h$

$2 h^{2}$

$2 a^{2}$

$0 a$

$0$

$3 a^{4}$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$0$

$3 a^{4}$

$0$

$8 a^{3} h$

$4 a^{2} h^{2}$

$0 a$

$8 a^{3} h$

$0$

$4 a^{2} h^{2}$

$0$

$4 h^{2} a^{2}$

$0$

Step 15

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

						$\frac{3 a^{2}}{2}$	-	$4 a h$	+	$2 h^{2}$
$2 a^{2}$	+	$0 a$	+	$0$		$3 a^{4}$	-	$8 a^{3} h$	+	$4 a^{2} h^{2}$	+	$0 a$	+	$0$
					-	$3 a^{4}$	-	$0$	-	$0$
							-	$8 a^{3} h$	+	$4 a^{2} h^{2}$	+	$0 a$
							+	$8 a^{3} h$	-	$0$	-	$0$
									+	$4 a^{2} h^{2}$	+	$0$	+	$0$
									-	$4 h^{2} a^{2}$	-	$0$	-	$0$

Step 16

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

						$\frac{3 a^{2}}{2}$	-	$4 a h$	+	$2 h^{2}$
$2 a^{2}$	+	$0 a$	+	$0$		$3 a^{4}$	-	$8 a^{3} h$	+	$4 a^{2} h^{2}$	+	$0 a$	+	$0$
					-	$3 a^{4}$	-	$0$	-	$0$
							-	$8 a^{3} h$	+	$4 a^{2} h^{2}$	+	$0 a$
							+	$8 a^{3} h$	-	$0$	-	$0$
									+	$4 a^{2} h^{2}$	+	$0$	+	$0$
									-	$4 h^{2} a^{2}$	-	$0$	-	$0$
														$0$

Step 17

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

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