Pre-Algebra Examples

$\frac{y^{3} + 9 y^{2} + 19 y - 6}{y + 5}$

Step 1

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

$y$

$5$

$y^{3}$

$9 y^{2}$

$19 y$

$6$

Step 2

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

					$y^{2}$
	$y$	+	$5$		$y^{3}$	+	$9 y^{2}$	+	$19 y$	-	$6$

Step 3

Multiply the new quotient term by the divisor.

				$y^{2}$
$y$	+	$5$		$y^{3}$	+	$9 y^{2}$	+	$19 y$	-	$6$
			+	$y^{3}$	+	$5 y^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $y^{3} + 5 y^{2}$

				$y^{2}$
$y$	+	$5$		$y^{3}$	+	$9 y^{2}$	+	$19 y$	-	$6$
			-	$y^{3}$	-	$5 y^{2}$

Step 5

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

				$y^{2}$
$y$	+	$5$		$y^{3}$	+	$9 y^{2}$	+	$19 y$	-	$6$
			-	$y^{3}$	-	$5 y^{2}$
					+	$4 y^{2}$

Step 6

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

				$y^{2}$
$y$	+	$5$		$y^{3}$	+	$9 y^{2}$	+	$19 y$	-	$6$
			-	$y^{3}$	-	$5 y^{2}$
					+	$4 y^{2}$	+	$19 y$

Step 7

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

				$y^{2}$	+	$4 y$
$y$	+	$5$		$y^{3}$	+	$9 y^{2}$	+	$19 y$	-	$6$
			-	$y^{3}$	-	$5 y^{2}$
					+	$4 y^{2}$	+	$19 y$

Step 8

Multiply the new quotient term by the divisor.

				$y^{2}$	+	$4 y$
$y$	+	$5$		$y^{3}$	+	$9 y^{2}$	+	$19 y$	-	$6$
			-	$y^{3}$	-	$5 y^{2}$
					+	$4 y^{2}$	+	$19 y$
					+	$4 y^{2}$	+	$20 y$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $4 y^{2} + 20 y$

				$y^{2}$	+	$4 y$
$y$	+	$5$		$y^{3}$	+	$9 y^{2}$	+	$19 y$	-	$6$
			-	$y^{3}$	-	$5 y^{2}$
					+	$4 y^{2}$	+	$19 y$
					-	$4 y^{2}$	-	$20 y$

Step 10

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

				$y^{2}$	+	$4 y$
$y$	+	$5$		$y^{3}$	+	$9 y^{2}$	+	$19 y$	-	$6$
			-	$y^{3}$	-	$5 y^{2}$
					+	$4 y^{2}$	+	$19 y$
					-	$4 y^{2}$	-	$20 y$
							-	$y$

Step 11

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

				$y^{2}$	+	$4 y$
$y$	+	$5$		$y^{3}$	+	$9 y^{2}$	+	$19 y$	-	$6$
			-	$y^{3}$	-	$5 y^{2}$
					+	$4 y^{2}$	+	$19 y$
					-	$4 y^{2}$	-	$20 y$
							-	$y$	-	$6$

Step 12

Divide the highest order term in the dividend $- y$ by the highest order term in divisor $y$ .

				$y^{2}$	+	$4 y$	-	$1$
$y$	+	$5$		$y^{3}$	+	$9 y^{2}$	+	$19 y$	-	$6$
			-	$y^{3}$	-	$5 y^{2}$
					+	$4 y^{2}$	+	$19 y$
					-	$4 y^{2}$	-	$20 y$
							-	$y$	-	$6$

Step 13

Multiply the new quotient term by the divisor.

				$y^{2}$	+	$4 y$	-	$1$
$y$	+	$5$		$y^{3}$	+	$9 y^{2}$	+	$19 y$	-	$6$
			-	$y^{3}$	-	$5 y^{2}$
					+	$4 y^{2}$	+	$19 y$
					-	$4 y^{2}$	-	$20 y$
							-	$y$	-	$6$
							-	$y$	-	$5$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- y - 5$

				$y^{2}$	+	$4 y$	-	$1$
$y$	+	$5$		$y^{3}$	+	$9 y^{2}$	+	$19 y$	-	$6$
			-	$y^{3}$	-	$5 y^{2}$
					+	$4 y^{2}$	+	$19 y$
					-	$4 y^{2}$	-	$20 y$
							-	$y$	-	$6$
							+	$y$	+	$5$

Step 15

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

				$y^{2}$	+	$4 y$	-	$1$
$y$	+	$5$		$y^{3}$	+	$9 y^{2}$	+	$19 y$	-	$6$
			-	$y^{3}$	-	$5 y^{2}$
					+	$4 y^{2}$	+	$19 y$
					-	$4 y^{2}$	-	$20 y$
							-	$y$	-	$6$
							+	$y$	+	$5$
									-	$1$

Step 16

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

$y^{2} + 4 y - 1 - \frac{1}{y + 5}$