Pre-Algebra Examples

$\frac{y^{3} + 3 y^{2} + 1}{y - 2}$

Step 1

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

$y$

$2$

$y^{3}$

$3 y^{2}$

$0 y$

$1$

Step 2

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

					$y^{2}$
	$y$	-	$2$		$y^{3}$	+	$3 y^{2}$	+	$0 y$	+	$1$

Step 3

Multiply the new quotient term by the divisor.

				$y^{2}$
$y$	-	$2$		$y^{3}$	+	$3 y^{2}$	+	$0 y$	+	$1$
			+	$y^{3}$	-	$2 y^{2}$

Step 4

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

				$y^{2}$
$y$	-	$2$		$y^{3}$	+	$3 y^{2}$	+	$0 y$	+	$1$
			-	$y^{3}$	+	$2 y^{2}$

Step 5

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

				$y^{2}$
$y$	-	$2$		$y^{3}$	+	$3 y^{2}$	+	$0 y$	+	$1$
			-	$y^{3}$	+	$2 y^{2}$
					+	$5 y^{2}$

Step 6

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

				$y^{2}$
$y$	-	$2$		$y^{3}$	+	$3 y^{2}$	+	$0 y$	+	$1$
			-	$y^{3}$	+	$2 y^{2}$
					+	$5 y^{2}$	+	$0 y$

Step 7

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

				$y^{2}$	+	$5 y$
$y$	-	$2$		$y^{3}$	+	$3 y^{2}$	+	$0 y$	+	$1$
			-	$y^{3}$	+	$2 y^{2}$
					+	$5 y^{2}$	+	$0 y$

Step 8

Multiply the new quotient term by the divisor.

				$y^{2}$	+	$5 y$
$y$	-	$2$		$y^{3}$	+	$3 y^{2}$	+	$0 y$	+	$1$
			-	$y^{3}$	+	$2 y^{2}$
					+	$5 y^{2}$	+	$0 y$
					+	$5 y^{2}$	-	$10 y$

Step 9

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

				$y^{2}$	+	$5 y$
$y$	-	$2$		$y^{3}$	+	$3 y^{2}$	+	$0 y$	+	$1$
			-	$y^{3}$	+	$2 y^{2}$
					+	$5 y^{2}$	+	$0 y$
					-	$5 y^{2}$	+	$10 y$

Step 10

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

				$y^{2}$	+	$5 y$
$y$	-	$2$		$y^{3}$	+	$3 y^{2}$	+	$0 y$	+	$1$
			-	$y^{3}$	+	$2 y^{2}$
					+	$5 y^{2}$	+	$0 y$
					-	$5 y^{2}$	+	$10 y$
							+	$10 y$

Step 11

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

				$y^{2}$	+	$5 y$
$y$	-	$2$		$y^{3}$	+	$3 y^{2}$	+	$0 y$	+	$1$
			-	$y^{3}$	+	$2 y^{2}$
					+	$5 y^{2}$	+	$0 y$
					-	$5 y^{2}$	+	$10 y$
							+	$10 y$	+	$1$

Step 12

Divide the highest order term in the dividend $10 y$ by the highest order term in divisor $y$ .

				$y^{2}$	+	$5 y$	+	$10$
$y$	-	$2$		$y^{3}$	+	$3 y^{2}$	+	$0 y$	+	$1$
			-	$y^{3}$	+	$2 y^{2}$
					+	$5 y^{2}$	+	$0 y$
					-	$5 y^{2}$	+	$10 y$
							+	$10 y$	+	$1$

Step 13

Multiply the new quotient term by the divisor.

				$y^{2}$	+	$5 y$	+	$10$
$y$	-	$2$		$y^{3}$	+	$3 y^{2}$	+	$0 y$	+	$1$
			-	$y^{3}$	+	$2 y^{2}$
					+	$5 y^{2}$	+	$0 y$
					-	$5 y^{2}$	+	$10 y$
							+	$10 y$	+	$1$
							+	$10 y$	-	$20$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $10 y - 20$

				$y^{2}$	+	$5 y$	+	$10$
$y$	-	$2$		$y^{3}$	+	$3 y^{2}$	+	$0 y$	+	$1$
			-	$y^{3}$	+	$2 y^{2}$
					+	$5 y^{2}$	+	$0 y$
					-	$5 y^{2}$	+	$10 y$
							+	$10 y$	+	$1$
							-	$10 y$	+	$20$

Step 15

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

				$y^{2}$	+	$5 y$	+	$10$
$y$	-	$2$		$y^{3}$	+	$3 y^{2}$	+	$0 y$	+	$1$
			-	$y^{3}$	+	$2 y^{2}$
					+	$5 y^{2}$	+	$0 y$
					-	$5 y^{2}$	+	$10 y$
							+	$10 y$	+	$1$
							-	$10 y$	+	$20$
									+	$21$

Step 16

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

$y^{2} + 5 y + 10 + \frac{21}{y - 2}$