Algebra Examples

$\frac{4 y^{4} + 8 y^{3} + 2 y^{2} - 8}{2 y + 4}$

Step 1

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

$2 y$

$4$

$4 y^{4}$

$8 y^{3}$

$2 y^{2}$

$0 y$

$8$

Step 2

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

$2 y^{3}$

$2 y$

$4$

$4 y^{4}$

$8 y^{3}$

$2 y^{2}$

$0 y$

$8$

Step 3

Multiply the new quotient term by the divisor.

$2 y^{3}$

$2 y$

$4$

$4 y^{4}$

$8 y^{3}$

$2 y^{2}$

$0 y$

$8$

$4 y^{4}$

$8 y^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $4 y^{4} + 8 y^{3}$

$2 y^{3}$

$2 y$

$4$

$4 y^{4}$

$8 y^{3}$

$2 y^{2}$

$0 y$

$8$

$4 y^{4}$

$8 y^{3}$

Step 5

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

$2 y^{3}$

$2 y$

$4$

$4 y^{4}$

$8 y^{3}$

$2 y^{2}$

$0 y$

$8$

$4 y^{4}$

$8 y^{3}$

$0$

Step 6

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

$2 y^{3}$

$2 y$

$4$

$4 y^{4}$

$8 y^{3}$

$2 y^{2}$

$0 y$

$8$

$4 y^{4}$

$8 y^{3}$

$0$

$2 y^{2}$

$0 y$

Step 7

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

$2 y^{3}$

$0 y^{2}$

$y$

$2 y$

$4$

$4 y^{4}$

$8 y^{3}$

$2 y^{2}$

$0 y$

$8$

$4 y^{4}$

$8 y^{3}$

$0$

$2 y^{2}$

$0 y$

Step 8

Multiply the new quotient term by the divisor.

$2 y^{3}$

$0 y^{2}$

$y$

$2 y$

$4$

$4 y^{4}$

$8 y^{3}$

$2 y^{2}$

$0 y$

$8$

$4 y^{4}$

$8 y^{3}$

$0$

$2 y^{2}$

$0 y$

$2 y^{2}$

$4 y$

Step 9

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

$2 y^{3}$

$0 y^{2}$

$y$

$2 y$

$4$

$4 y^{4}$

$8 y^{3}$

$2 y^{2}$

$0 y$

$8$

$4 y^{4}$

$8 y^{3}$

$0$

$2 y^{2}$

$0 y$

$2 y^{2}$

$4 y$

Step 10

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

$2 y^{3}$

$0 y^{2}$

$y$

$2 y$

$4$

$4 y^{4}$

$8 y^{3}$

$2 y^{2}$

$0 y$

$8$

$4 y^{4}$

$8 y^{3}$

$0$

$2 y^{2}$

$0 y$

$2 y^{2}$

$4 y$

Step 11

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

$2 y^{3}$

$0 y^{2}$

$y$

$2 y$

$4$

$4 y^{4}$

$8 y^{3}$

$2 y^{2}$

$0 y$

$8$

$4 y^{4}$

$8 y^{3}$

$0$

$2 y^{2}$

$0 y$

$2 y^{2}$

$4 y$

$8$

Step 12

Divide the highest order term in the dividend $- 4 y$ by the highest order term in divisor $2 y$ .

$2 y^{3}$

$0 y^{2}$

$y$

$2$

$2 y$

$4$

$4 y^{4}$

$8 y^{3}$

$2 y^{2}$

$0 y$

$8$

$4 y^{4}$

$8 y^{3}$

$0$

$2 y^{2}$

$0 y$

$2 y^{2}$

$4 y$

$8$

Step 13

Multiply the new quotient term by the divisor.

				$2 y^{3}$	+	$0 y^{2}$	+	$y$	-	$2$
$2 y$	+	$4$		$4 y^{4}$	+	$8 y^{3}$	+	$2 y^{2}$	+	$0 y$	-	$8$
			-	$4 y^{4}$	-	$8 y^{3}$
						$0$	+	$2 y^{2}$	+	$0 y$
							-	$2 y^{2}$	-	$4 y$
									-	$4 y$	-	$8$
									-	$4 y$	-	$8$

Step 14

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

$2 y^{3}$

$0 y^{2}$

$y$

$2$

$2 y$

$4$

$4 y^{4}$

$8 y^{3}$

$2 y^{2}$

$0 y$

$8$

$4 y^{4}$

$8 y^{3}$

$0$

$2 y^{2}$

$0 y$

$2 y^{2}$

$4 y$

$8$

$4 y$

$8$

Step 15

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

$2 y^{3}$

$0 y^{2}$

$y$

$2$

$2 y$

$4$

$4 y^{4}$

$8 y^{3}$

$2 y^{2}$

$0 y$

$8$

$4 y^{4}$

$8 y^{3}$

$0$

$2 y^{2}$

$0 y$

$2 y^{2}$

$4 y$

$8$

$4 y$

$8$

$0$

Step 16

Since the remander is $0$ , the final answer is the quotient.

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