Pre-Algebra Examples

$(8 v^{3} + 31 v^{2} - 11^{2} - 34) \div (v + 4)$

Step 1

Expand $8 v^{3} + 31 v^{2} - 11^{2} - 34$ .

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Step 1.1

Rewrite the exponentiation as a product.

$\frac{8 v^{3} + 31 v^{2} - (11 \cdot 11) - 34}{v + 4}$

Step 1.2

Remove parentheses.

$\frac{8 v^{3} + 31 v^{2} - 1 \cdot 11 \cdot 11 - 34}{v + 4}$

Step 1.3

Multiply $- 1$ by $11$ .

$\frac{8 v^{3} + 31 v^{2} - 11 \cdot 11 - 34}{v + 4}$

Step 1.4

Multiply $- 11$ by $11$ .

$\frac{8 v^{3} + 31 v^{2} - 121 - 34}{v + 4}$

Step 1.5

Subtract $34$ from $- 121$ .

$\frac{8 v^{3} + 31 v^{2} - 155}{v + 4}$

Step 2

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

$v$

$4$

$8 v^{3}$

$31 v^{2}$

$0 v$

$155$

Step 3

Divide the highest order term in the dividend $8 v^{3}$ by the highest order term in divisor $v$ .

					$8 v^{2}$
	$v$	+	$4$		$8 v^{3}$	+	$31 v^{2}$	+	$0 v$	-	$155$

Step 4

Multiply the new quotient term by the divisor.

				$8 v^{2}$
$v$	+	$4$		$8 v^{3}$	+	$31 v^{2}$	+	$0 v$	-	$155$
			+	$8 v^{3}$	+	$32 v^{2}$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $8 v^{3} + 32 v^{2}$

				$8 v^{2}$
$v$	+	$4$		$8 v^{3}$	+	$31 v^{2}$	+	$0 v$	-	$155$
			-	$8 v^{3}$	-	$32 v^{2}$

Step 6

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

				$8 v^{2}$
$v$	+	$4$		$8 v^{3}$	+	$31 v^{2}$	+	$0 v$	-	$155$
			-	$8 v^{3}$	-	$32 v^{2}$
					-	$v^{2}$

Step 7

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

				$8 v^{2}$
$v$	+	$4$		$8 v^{3}$	+	$31 v^{2}$	+	$0 v$	-	$155$
			-	$8 v^{3}$	-	$32 v^{2}$
					-	$v^{2}$	+	$0 v$

Step 8

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

				$8 v^{2}$	-	$v$
$v$	+	$4$		$8 v^{3}$	+	$31 v^{2}$	+	$0 v$	-	$155$
			-	$8 v^{3}$	-	$32 v^{2}$
					-	$v^{2}$	+	$0 v$

Step 9

Multiply the new quotient term by the divisor.

				$8 v^{2}$	-	$v$
$v$	+	$4$		$8 v^{3}$	+	$31 v^{2}$	+	$0 v$	-	$155$
			-	$8 v^{3}$	-	$32 v^{2}$
					-	$v^{2}$	+	$0 v$
					-	$v^{2}$	-	$4 v$

Step 10

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

				$8 v^{2}$	-	$v$
$v$	+	$4$		$8 v^{3}$	+	$31 v^{2}$	+	$0 v$	-	$155$
			-	$8 v^{3}$	-	$32 v^{2}$
					-	$v^{2}$	+	$0 v$
					+	$v^{2}$	+	$4 v$

Step 11

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

				$8 v^{2}$	-	$v$
$v$	+	$4$		$8 v^{3}$	+	$31 v^{2}$	+	$0 v$	-	$155$
			-	$8 v^{3}$	-	$32 v^{2}$
					-	$v^{2}$	+	$0 v$
					+	$v^{2}$	+	$4 v$
							+	$4 v$

Step 12

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

				$8 v^{2}$	-	$v$
$v$	+	$4$		$8 v^{3}$	+	$31 v^{2}$	+	$0 v$	-	$155$
			-	$8 v^{3}$	-	$32 v^{2}$
					-	$v^{2}$	+	$0 v$
					+	$v^{2}$	+	$4 v$
							+	$4 v$	-	$155$

Step 13

Divide the highest order term in the dividend $4 v$ by the highest order term in divisor $v$ .

				$8 v^{2}$	-	$v$	+	$4$
$v$	+	$4$		$8 v^{3}$	+	$31 v^{2}$	+	$0 v$	-	$155$
			-	$8 v^{3}$	-	$32 v^{2}$
					-	$v^{2}$	+	$0 v$
					+	$v^{2}$	+	$4 v$
							+	$4 v$	-	$155$

Step 14

Multiply the new quotient term by the divisor.

				$8 v^{2}$	-	$v$	+	$4$
$v$	+	$4$		$8 v^{3}$	+	$31 v^{2}$	+	$0 v$	-	$155$
			-	$8 v^{3}$	-	$32 v^{2}$
					-	$v^{2}$	+	$0 v$
					+	$v^{2}$	+	$4 v$
							+	$4 v$	-	$155$
							+	$4 v$	+	$16$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $4 v + 16$

				$8 v^{2}$	-	$v$	+	$4$
$v$	+	$4$		$8 v^{3}$	+	$31 v^{2}$	+	$0 v$	-	$155$
			-	$8 v^{3}$	-	$32 v^{2}$
					-	$v^{2}$	+	$0 v$
					+	$v^{2}$	+	$4 v$
							+	$4 v$	-	$155$
							-	$4 v$	-	$16$

Step 16

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

				$8 v^{2}$	-	$v$	+	$4$
$v$	+	$4$		$8 v^{3}$	+	$31 v^{2}$	+	$0 v$	-	$155$
			-	$8 v^{3}$	-	$32 v^{2}$
					-	$v^{2}$	+	$0 v$
					+	$v^{2}$	+	$4 v$
							+	$4 v$	-	$155$
							-	$4 v$	-	$16$
									-	$171$

Step 17

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

$8 v^{2} - v + 4 - \frac{171}{v + 4}$