Pre-Algebra Examples

$(n^{3} + 7 n^{2} + 14 n + 3) \div (n + 2)$

Step 1

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

$n$

$2$

$n^{3}$

$7 n^{2}$

$14 n$

$3$

Step 2

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

					$n^{2}$
	$n$	+	$2$		$n^{3}$	+	$7 n^{2}$	+	$14 n$	+	$3$

Step 3

Multiply the new quotient term by the divisor.

				$n^{2}$
$n$	+	$2$		$n^{3}$	+	$7 n^{2}$	+	$14 n$	+	$3$
			+	$n^{3}$	+	$2 n^{2}$

Step 4

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

				$n^{2}$
$n$	+	$2$		$n^{3}$	+	$7 n^{2}$	+	$14 n$	+	$3$
			-	$n^{3}$	-	$2 n^{2}$

Step 5

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

				$n^{2}$
$n$	+	$2$		$n^{3}$	+	$7 n^{2}$	+	$14 n$	+	$3$
			-	$n^{3}$	-	$2 n^{2}$
					+	$5 n^{2}$

Step 6

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

				$n^{2}$
$n$	+	$2$		$n^{3}$	+	$7 n^{2}$	+	$14 n$	+	$3$
			-	$n^{3}$	-	$2 n^{2}$
					+	$5 n^{2}$	+	$14 n$

Step 7

Divide the highest order term in the dividend $5 n^{2}$ by the highest order term in divisor $n$ .

				$n^{2}$	+	$5 n$
$n$	+	$2$		$n^{3}$	+	$7 n^{2}$	+	$14 n$	+	$3$
			-	$n^{3}$	-	$2 n^{2}$
					+	$5 n^{2}$	+	$14 n$

Step 8

Multiply the new quotient term by the divisor.

				$n^{2}$	+	$5 n$
$n$	+	$2$		$n^{3}$	+	$7 n^{2}$	+	$14 n$	+	$3$
			-	$n^{3}$	-	$2 n^{2}$
					+	$5 n^{2}$	+	$14 n$
					+	$5 n^{2}$	+	$10 n$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $5 n^{2} + 10 n$

				$n^{2}$	+	$5 n$
$n$	+	$2$		$n^{3}$	+	$7 n^{2}$	+	$14 n$	+	$3$
			-	$n^{3}$	-	$2 n^{2}$
					+	$5 n^{2}$	+	$14 n$
					-	$5 n^{2}$	-	$10 n$

Step 10

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

				$n^{2}$	+	$5 n$
$n$	+	$2$		$n^{3}$	+	$7 n^{2}$	+	$14 n$	+	$3$
			-	$n^{3}$	-	$2 n^{2}$
					+	$5 n^{2}$	+	$14 n$
					-	$5 n^{2}$	-	$10 n$
							+	$4 n$

Step 11

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

				$n^{2}$	+	$5 n$
$n$	+	$2$		$n^{3}$	+	$7 n^{2}$	+	$14 n$	+	$3$
			-	$n^{3}$	-	$2 n^{2}$
					+	$5 n^{2}$	+	$14 n$
					-	$5 n^{2}$	-	$10 n$
							+	$4 n$	+	$3$

Step 12

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

				$n^{2}$	+	$5 n$	+	$4$
$n$	+	$2$		$n^{3}$	+	$7 n^{2}$	+	$14 n$	+	$3$
			-	$n^{3}$	-	$2 n^{2}$
					+	$5 n^{2}$	+	$14 n$
					-	$5 n^{2}$	-	$10 n$
							+	$4 n$	+	$3$

Step 13

Multiply the new quotient term by the divisor.

				$n^{2}$	+	$5 n$	+	$4$
$n$	+	$2$		$n^{3}$	+	$7 n^{2}$	+	$14 n$	+	$3$
			-	$n^{3}$	-	$2 n^{2}$
					+	$5 n^{2}$	+	$14 n$
					-	$5 n^{2}$	-	$10 n$
							+	$4 n$	+	$3$
							+	$4 n$	+	$8$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $4 n + 8$

				$n^{2}$	+	$5 n$	+	$4$
$n$	+	$2$		$n^{3}$	+	$7 n^{2}$	+	$14 n$	+	$3$
			-	$n^{3}$	-	$2 n^{2}$
					+	$5 n^{2}$	+	$14 n$
					-	$5 n^{2}$	-	$10 n$
							+	$4 n$	+	$3$
							-	$4 n$	-	$8$

Step 15

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

				$n^{2}$	+	$5 n$	+	$4$
$n$	+	$2$		$n^{3}$	+	$7 n^{2}$	+	$14 n$	+	$3$
			-	$n^{3}$	-	$2 n^{2}$
					+	$5 n^{2}$	+	$14 n$
					-	$5 n^{2}$	-	$10 n$
							+	$4 n$	+	$3$
							-	$4 n$	-	$8$
									-	$5$

Step 16

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

$n^{2} + 5 n + 4 - \frac{5}{n + 2}$