Pre-Algebra Examples

$(42 w^{3} + 9 w^{2} \cdot (48 w) + 34) \div (6 w + 3)$

Step 1

Expand $42 w^{3} + 9 w^{2} \cdot (48 w) + 34$ .

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Step 1.1

Remove parentheses.

$\frac{42 w^{3} + 9 w^{2} \cdot (48 w) + 34}{6 w + 3}$

Step 1.2

Move $w^{2}$ .

$\frac{42 w^{3} + 9 \cdot (48 w^{2} w) + 34}{6 w + 3}$

Step 1.3

Multiply $9$ by $48$ .

$\frac{42 w^{3} + 432 w^{2} w + 34}{6 w + 3}$

Step 1.4

Raise $w$ to the power of $1$ .

$\frac{42 w^{3} + 432 (w^{2} w) + 34}{6 w + 3}$

Step 1.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{42 w^{3} + 432 w^{2 + 1} + 34}{6 w + 3}$

Step 1.6

Add $2$ and $1$ .

$\frac{42 w^{3} + 432 w^{3} + 34}{6 w + 3}$

Step 1.7

Add $42 w^{3}$ and $432 w^{3}$ .

$\frac{474 w^{3} + 34}{6 w + 3}$

Step 2

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

$6 w$

$3$

$474 w^{3}$

$0 w^{2}$

$0 w$

$34$

Step 3

Divide the highest order term in the dividend $474 w^{3}$ by the highest order term in divisor $6 w$ .

					$79 w^{2}$
	$6 w$	+	$3$		$474 w^{3}$	+	$0 w^{2}$	+	$0 w$	+	$34$

Step 4

Multiply the new quotient term by the divisor.

				$79 w^{2}$
$6 w$	+	$3$		$474 w^{3}$	+	$0 w^{2}$	+	$0 w$	+	$34$
			+	$474 w^{3}$	+	$237 w^{2}$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $474 w^{3} + 237 w^{2}$

				$79 w^{2}$
$6 w$	+	$3$		$474 w^{3}$	+	$0 w^{2}$	+	$0 w$	+	$34$
			-	$474 w^{3}$	-	$237 w^{2}$

Step 6

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

				$79 w^{2}$
$6 w$	+	$3$		$474 w^{3}$	+	$0 w^{2}$	+	$0 w$	+	$34$
			-	$474 w^{3}$	-	$237 w^{2}$
					-	$237 w^{2}$

Step 7

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

				$79 w^{2}$
$6 w$	+	$3$		$474 w^{3}$	+	$0 w^{2}$	+	$0 w$	+	$34$
			-	$474 w^{3}$	-	$237 w^{2}$
					-	$237 w^{2}$	+	$0 w$

Step 8

Divide the highest order term in the dividend $- 237 w^{2}$ by the highest order term in divisor $6 w$ .

				$79 w^{2}$	-	$\frac{79 w}{2}$
$6 w$	+	$3$		$474 w^{3}$	+	$0 w^{2}$	+	$0 w$	+	$34$
			-	$474 w^{3}$	-	$237 w^{2}$
					-	$237 w^{2}$	+	$0 w$

Step 9

Multiply the new quotient term by the divisor.

				$79 w^{2}$	-	$\frac{79 w}{2}$
$6 w$	+	$3$		$474 w^{3}$	+	$0 w^{2}$	+	$0 w$	+	$34$
			-	$474 w^{3}$	-	$237 w^{2}$
					-	$237 w^{2}$	+	$0 w$
					-	$237 w^{2}$	-	$\frac{237 w}{2}$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $- 237 w^{2} - \frac{237 w}{2}$

				$79 w^{2}$	-	$\frac{79 w}{2}$
$6 w$	+	$3$		$474 w^{3}$	+	$0 w^{2}$	+	$0 w$	+	$34$
			-	$474 w^{3}$	-	$237 w^{2}$
					-	$237 w^{2}$	+	$0 w$
					+	$237 w^{2}$	+	$\frac{237 w}{2}$

Step 11

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

				$79 w^{2}$	-	$\frac{79 w}{2}$
$6 w$	+	$3$		$474 w^{3}$	+	$0 w^{2}$	+	$0 w$	+	$34$
			-	$474 w^{3}$	-	$237 w^{2}$
					-	$237 w^{2}$	+	$0 w$
					+	$237 w^{2}$	+	$\frac{237 w}{2}$
							+	$\frac{237 w}{2}$

Step 12

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

				$79 w^{2}$	-	$\frac{79 w}{2}$
$6 w$	+	$3$		$474 w^{3}$	+	$0 w^{2}$	+	$0 w$	+	$34$
			-	$474 w^{3}$	-	$237 w^{2}$
					-	$237 w^{2}$	+	$0 w$
					+	$237 w^{2}$	+	$\frac{237 w}{2}$
							+	$\frac{237 w}{2}$	+	$34$

Step 13

Divide the highest order term in the dividend $\frac{237 w}{2}$ by the highest order term in divisor $6 w$ .

				$79 w^{2}$	-	$\frac{79 w}{2}$	+	$\frac{79}{4}$
$6 w$	+	$3$		$474 w^{3}$	+	$0 w^{2}$	+	$0 w$	+	$34$
			-	$474 w^{3}$	-	$237 w^{2}$
					-	$237 w^{2}$	+	$0 w$
					+	$237 w^{2}$	+	$\frac{237 w}{2}$
							+	$\frac{237 w}{2}$	+	$34$

Step 14

Multiply the new quotient term by the divisor.

				$79 w^{2}$	-	$\frac{79 w}{2}$	+	$\frac{79}{4}$
$6 w$	+	$3$		$474 w^{3}$	+	$0 w^{2}$	+	$0 w$	+	$34$
			-	$474 w^{3}$	-	$237 w^{2}$
					-	$237 w^{2}$	+	$0 w$
					+	$237 w^{2}$	+	$\frac{237 w}{2}$
							+	$\frac{237 w}{2}$	+	$34$
							+	$\frac{237 w}{2}$	+	$\frac{237}{4}$

Step 15

The expression needs to be subtracted from the dividend, so change all the signs in $\frac{237 w}{2} + \frac{237}{4}$

				$79 w^{2}$	-	$\frac{79 w}{2}$	+	$\frac{79}{4}$
$6 w$	+	$3$		$474 w^{3}$	+	$0 w^{2}$	+	$0 w$	+	$34$
			-	$474 w^{3}$	-	$237 w^{2}$
					-	$237 w^{2}$	+	$0 w$
					+	$237 w^{2}$	+	$\frac{237 w}{2}$
							+	$\frac{237 w}{2}$	+	$34$
							-	$\frac{237 w}{2}$	-	$\frac{237}{4}$

Step 16

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

				$79 w^{2}$	-	$\frac{79 w}{2}$	+	$\frac{79}{4}$
$6 w$	+	$3$		$474 w^{3}$	+	$0 w^{2}$	+	$0 w$	+	$34$
			-	$474 w^{3}$	-	$237 w^{2}$
					-	$237 w^{2}$	+	$0 w$
					+	$237 w^{2}$	+	$\frac{237 w}{2}$
							+	$\frac{237 w}{2}$	+	$34$
							-	$\frac{237 w}{2}$	-	$\frac{237}{4}$
									-	$\frac{101}{4}$

Step 17

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

$79 w^{2} - \frac{79 w}{2} + \frac{79}{4} - \frac{101}{4 (6 w + 3)}$