Algebra Examples

$\frac{- 8 x^{4} - 5 x^{3}}{x^{2} - 3}$

Step 1

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

$x^{2}$

$0 x$

$3$

$8 x^{4}$

$5 x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 2

Divide the highest order term in the dividend $- 8 x^{4}$ by the highest order term in divisor $x^{2}$ .

$8 x^{2}$

$x^{2}$

$0 x$

$3$

$8 x^{4}$

$5 x^{3}$

$0 x^{2}$

$0 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$8 x^{2}$

$x^{2}$

$0 x$

$3$

$8 x^{4}$

$5 x^{3}$

$0 x^{2}$

$0 x$

$0$

$8 x^{4}$

$0$

$24 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{4} + 0 + 24 x^{2}$

$8 x^{2}$

$x^{2}$

$0 x$

$3$

$8 x^{4}$

$5 x^{3}$

$0 x^{2}$

$0 x$

$0$

$8 x^{4}$

$0$

$24 x^{2}$

Step 5

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

					-	$8 x^{2}$
$x^{2}$	+	$0 x$	-	$3$	-	$8 x^{4}$	-	$5 x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$0$
					+	$8 x^{4}$	-	$0$	-	$24 x^{2}$
							-	$5 x^{3}$	-	$24 x^{2}$

Step 6

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

$8 x^{2}$

$x^{2}$

$0 x$

$3$

$8 x^{4}$

$5 x^{3}$

$0 x^{2}$

$0 x$

$0$

$8 x^{4}$

$0$

$24 x^{2}$

$5 x^{3}$

$24 x^{2}$

$0 x$

Step 7

Divide the highest order term in the dividend $- 5 x^{3}$ by the highest order term in divisor $x^{2}$ .

$8 x^{2}$

$5 x$

$x^{2}$

$0 x$

$3$

$8 x^{4}$

$5 x^{3}$

$0 x^{2}$

$0 x$

$0$

$8 x^{4}$

$0$

$24 x^{2}$

$5 x^{3}$

$24 x^{2}$

$0 x$

Step 8

Multiply the new quotient term by the divisor.

$8 x^{2}$

$5 x$

$x^{2}$

$0 x$

$3$

$8 x^{4}$

$5 x^{3}$

$0 x^{2}$

$0 x$

$0$

$8 x^{4}$

$0$

$24 x^{2}$

$5 x^{3}$

$24 x^{2}$

$0 x$

$5 x^{3}$

$0$

$15 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{3} + 0 + 15 x$

					-	$8 x^{2}$	-	$5 x$
$x^{2}$	+	$0 x$	-	$3$	-	$8 x^{4}$	-	$5 x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$0$
					+	$8 x^{4}$	-	$0$	-	$24 x^{2}$
							-	$5 x^{3}$	-	$24 x^{2}$	+	$0 x$
							+	$5 x^{3}$	-	$0$	-	$15 x$

Step 10

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

					-	$8 x^{2}$	-	$5 x$
$x^{2}$	+	$0 x$	-	$3$	-	$8 x^{4}$	-	$5 x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$0$
					+	$8 x^{4}$	-	$0$	-	$24 x^{2}$
							-	$5 x^{3}$	-	$24 x^{2}$	+	$0 x$
							+	$5 x^{3}$	-	$0$	-	$15 x$
									-	$24 x^{2}$	-	$15 x$

Step 11

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

					-	$8 x^{2}$	-	$5 x$
$x^{2}$	+	$0 x$	-	$3$	-	$8 x^{4}$	-	$5 x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$0$
					+	$8 x^{4}$	-	$0$	-	$24 x^{2}$
							-	$5 x^{3}$	-	$24 x^{2}$	+	$0 x$
							+	$5 x^{3}$	-	$0$	-	$15 x$
									-	$24 x^{2}$	-	$15 x$	+	$0$

Step 12

Divide the highest order term in the dividend $- 24 x^{2}$ by the highest order term in divisor $x^{2}$ .

					-	$8 x^{2}$	-	$5 x$	-	$24$
$x^{2}$	+	$0 x$	-	$3$	-	$8 x^{4}$	-	$5 x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$0$
					+	$8 x^{4}$	-	$0$	-	$24 x^{2}$
							-	$5 x^{3}$	-	$24 x^{2}$	+	$0 x$
							+	$5 x^{3}$	-	$0$	-	$15 x$
									-	$24 x^{2}$	-	$15 x$	+	$0$

Step 13

Multiply the new quotient term by the divisor.

					-	$8 x^{2}$	-	$5 x$	-	$24$
$x^{2}$	+	$0 x$	-	$3$	-	$8 x^{4}$	-	$5 x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$0$
					+	$8 x^{4}$	-	$0$	-	$24 x^{2}$
							-	$5 x^{3}$	-	$24 x^{2}$	+	$0 x$
							+	$5 x^{3}$	-	$0$	-	$15 x$
									-	$24 x^{2}$	-	$15 x$	+	$0$
									-	$24 x^{2}$	+	$0$	+	$72$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 24 x^{2} + 0 + 72$

					-	$8 x^{2}$	-	$5 x$	-	$24$
$x^{2}$	+	$0 x$	-	$3$	-	$8 x^{4}$	-	$5 x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$0$
					+	$8 x^{4}$	-	$0$	-	$24 x^{2}$
							-	$5 x^{3}$	-	$24 x^{2}$	+	$0 x$
							+	$5 x^{3}$	-	$0$	-	$15 x$
									-	$24 x^{2}$	-	$15 x$	+	$0$
									+	$24 x^{2}$	-	$0$	-	$72$

Step 15

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

					-	$8 x^{2}$	-	$5 x$	-	$24$
$x^{2}$	+	$0 x$	-	$3$	-	$8 x^{4}$	-	$5 x^{3}$	+	$0 x^{2}$	+	$0 x$	+	$0$
					+	$8 x^{4}$	-	$0$	-	$24 x^{2}$
							-	$5 x^{3}$	-	$24 x^{2}$	+	$0 x$
							+	$5 x^{3}$	-	$0$	-	$15 x$
									-	$24 x^{2}$	-	$15 x$	+	$0$
									+	$24 x^{2}$	-	$0$	-	$72$
											-	$15 x$	-	$72$

Step 16

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

$- 8 x^{2} - 5 x - 24 + \frac{- 15 x - 72}{x^{2} - 3}$