Algebra Examples

$\frac{2 x^{6} - 9 x^{5} + 4 x^{2} - 5}{x^{3} - 5}$

Step 1

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

$x^{3}$

$0 x^{2}$

$0 x$

$5$

$2 x^{6}$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$5$

Step 2

Divide the highest order term in the dividend $2 x^{6}$ by the highest order term in divisor $x^{3}$ .

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$0 x$

$5$

$2 x^{6}$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$5$

Step 3

Multiply the new quotient term by the divisor.

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$0 x$

$5$

$2 x^{6}$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$5$

$2 x^{6}$

$0$

$10 x^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{6} + 0 + 0 - 10 x^{3}$

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$0 x$

$5$

$2 x^{6}$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$5$

$2 x^{6}$

$0$

$10 x^{3}$

Step 5

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

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$0 x$

$5$

$2 x^{6}$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$5$

$2 x^{6}$

$0$

$10 x^{3}$

$9 x^{5}$

$0$

$10 x^{3}$

Step 6

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

$2 x^{3}$

$x^{3}$

$0 x^{2}$

$0 x$

$5$

$2 x^{6}$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$5$

$2 x^{6}$

$0$

$10 x^{3}$

$9 x^{5}$

$0$

$10 x^{3}$

$4 x^{2}$

$0 x$

Step 7

Divide the highest order term in the dividend $- 9 x^{5}$ by the highest order term in divisor $x^{3}$ .

$2 x^{3}$

$9 x^{2}$

$x^{3}$

$0 x^{2}$

$0 x$

$5$

$2 x^{6}$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$5$

$2 x^{6}$

$0$

$10 x^{3}$

$9 x^{5}$

$0$

$10 x^{3}$

$4 x^{2}$

$0 x$

Step 8

Multiply the new quotient term by the divisor.

$2 x^{3}$

$9 x^{2}$

$x^{3}$

$0 x^{2}$

$0 x$

$5$

$2 x^{6}$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$5$

$2 x^{6}$

$0$

$10 x^{3}$

$9 x^{5}$

$0$

$10 x^{3}$

$4 x^{2}$

$0 x$

$9 x^{5}$

$0$

$45 x^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 9 x^{5} + 0 + 0 + 45 x^{2}$

$2 x^{3}$

$9 x^{2}$

$x^{3}$

$0 x^{2}$

$0 x$

$5$

$2 x^{6}$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$5$

$2 x^{6}$

$0$

$10 x^{3}$

$9 x^{5}$

$0$

$10 x^{3}$

$4 x^{2}$

$0 x$

$9 x^{5}$

$0$

$45 x^{2}$

Step 10

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

$2 x^{3}$

$9 x^{2}$

$x^{3}$

$0 x^{2}$

$0 x$

$5$

$2 x^{6}$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$5$

$2 x^{6}$

$0$

$10 x^{3}$

$9 x^{5}$

$0$

$10 x^{3}$

$4 x^{2}$

$0 x$

$9 x^{5}$

$0$

$45 x^{2}$

$10 x^{3}$

$41 x^{2}$

Step 11

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

$2 x^{3}$

$9 x^{2}$

$x^{3}$

$0 x^{2}$

$0 x$

$5$

$2 x^{6}$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$5$

$2 x^{6}$

$0$

$10 x^{3}$

$9 x^{5}$

$0$

$10 x^{3}$

$4 x^{2}$

$0 x$

$9 x^{5}$

$0$

$45 x^{2}$

$10 x^{3}$

$41 x^{2}$

$0 x$

$5$

Step 12

Divide the highest order term in the dividend $10 x^{3}$ by the highest order term in divisor $x^{3}$ .

$2 x^{3}$

$9 x^{2}$

$0 x$

$10$

$x^{3}$

$0 x^{2}$

$0 x$

$5$

$2 x^{6}$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$5$

$2 x^{6}$

$0$

$10 x^{3}$

$9 x^{5}$

$0$

$10 x^{3}$

$4 x^{2}$

$0 x$

$9 x^{5}$

$0$

$45 x^{2}$

$10 x^{3}$

$41 x^{2}$

$0 x$

$5$

Step 13

Multiply the new quotient term by the divisor.

$2 x^{3}$

$9 x^{2}$

$0 x$

$10$

$x^{3}$

$0 x^{2}$

$0 x$

$5$

$2 x^{6}$

$9 x^{5}$

$0 x^{4}$

$0 x^{3}$

$4 x^{2}$

$0 x$

$5$

$2 x^{6}$

$0$

$10 x^{3}$

$9 x^{5}$

$0$

$10 x^{3}$

$4 x^{2}$

$0 x$

$9 x^{5}$

$0$

$45 x^{2}$

$10 x^{3}$

$41 x^{2}$

$0 x$

$5$

$10 x^{3}$

$0$

$50$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{3} + 0 + 0 - 50$

								$2 x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$10$
$x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$5$		$2 x^{6}$	-	$9 x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$5$
							-	$2 x^{6}$	-	$0$	-	$0$	+	$10 x^{3}$
									-	$9 x^{5}$	+	$0$	+	$10 x^{3}$	+	$4 x^{2}$	+	$0 x$
									+	$9 x^{5}$	-	$0$	-	$0$	-	$45 x^{2}$
													+	$10 x^{3}$	-	$41 x^{2}$	+	$0 x$	-	$5$
													-	$10 x^{3}$	-	$0$	-	$0$	+	$50$

Step 15

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

								$2 x^{3}$	-	$9 x^{2}$	+	$0 x$	+	$10$
$x^{3}$	+	$0 x^{2}$	+	$0 x$	-	$5$		$2 x^{6}$	-	$9 x^{5}$	+	$0 x^{4}$	+	$0 x^{3}$	+	$4 x^{2}$	+	$0 x$	-	$5$
							-	$2 x^{6}$	-	$0$	-	$0$	+	$10 x^{3}$
									-	$9 x^{5}$	+	$0$	+	$10 x^{3}$	+	$4 x^{2}$	+	$0 x$
									+	$9 x^{5}$	-	$0$	-	$0$	-	$45 x^{2}$
													+	$10 x^{3}$	-	$41 x^{2}$	+	$0 x$	-	$5$
													-	$10 x^{3}$	-	$0$	-	$0$	+	$50$
															-	$41 x^{2}$	+	$0$	+	$45$

Step 16

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

$2 x^{3} - 9 x^{2} + 10 + \frac{- 41 x^{2} + 45}{x^{3} - 5}$