Algebra Examples

$(x^{5} - 3 x^{3} - 3 x^{2} - 10 x + 15) \div (x^{2} - 5)$

Step 1

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

$x^{2}$

$0 x$

$5$

$x^{5}$

$0 x^{4}$

$3 x^{3}$

$3 x^{2}$

$10 x$

$15$

Step 2

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

$x^{3}$

$x^{2}$

$0 x$

$5$

$x^{5}$

$0 x^{4}$

$3 x^{3}$

$3 x^{2}$

$10 x$

$15$

Step 3

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$0 x$

$5$

$x^{5}$

$0 x^{4}$

$3 x^{3}$

$3 x^{2}$

$10 x$

$15$

$x^{5}$

$0$

$5 x^{3}$

Step 4

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

$x^{3}$

$x^{2}$

$0 x$

$5$

$x^{5}$

$0 x^{4}$

$3 x^{3}$

$3 x^{2}$

$10 x$

$15$

$x^{5}$

$0$

$5 x^{3}$

Step 5

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

$x^{3}$

$x^{2}$

$0 x$

$5$

$x^{5}$

$0 x^{4}$

$3 x^{3}$

$3 x^{2}$

$10 x$

$15$

$x^{5}$

$0$

$5 x^{3}$

$2 x^{3}$

Step 6

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

$x^{3}$

$x^{2}$

$0 x$

$5$

$x^{5}$

$0 x^{4}$

$3 x^{3}$

$3 x^{2}$

$10 x$

$15$

$x^{5}$

$0$

$5 x^{3}$

$2 x^{3}$

$3 x^{2}$

$10 x$

Step 7

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

$x^{3}$

$0 x^{2}$

$2 x$

$x^{2}$

$0 x$

$5$

$x^{5}$

$0 x^{4}$

$3 x^{3}$

$3 x^{2}$

$10 x$

$15$

$x^{5}$

$0$

$5 x^{3}$

$2 x^{3}$

$3 x^{2}$

$10 x$

Step 8

Multiply the new quotient term by the divisor.

$x^{3}$

$0 x^{2}$

$2 x$

$x^{2}$

$0 x$

$5$

$x^{5}$

$0 x^{4}$

$3 x^{3}$

$3 x^{2}$

$10 x$

$15$

$x^{5}$

$0$

$5 x^{3}$

$2 x^{3}$

$3 x^{2}$

$10 x$

$2 x^{3}$

$0$

$10 x$

Step 9

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

$x^{3}$

$0 x^{2}$

$2 x$

$x^{2}$

$0 x$

$5$

$x^{5}$

$0 x^{4}$

$3 x^{3}$

$3 x^{2}$

$10 x$

$15$

$x^{5}$

$0$

$5 x^{3}$

$2 x^{3}$

$3 x^{2}$

$10 x$

$2 x^{3}$

$0$

$10 x$

Step 10

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

$x^{3}$

$0 x^{2}$

$2 x$

$x^{2}$

$0 x$

$5$

$x^{5}$

$0 x^{4}$

$3 x^{3}$

$3 x^{2}$

$10 x$

$15$

$x^{5}$

$0$

$5 x^{3}$

$2 x^{3}$

$3 x^{2}$

$10 x$

$2 x^{3}$

$0$

$10 x$

$3 x^{2}$

$0$

Step 11

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

$x^{3}$

$0 x^{2}$

$2 x$

$x^{2}$

$0 x$

$5$

$x^{5}$

$0 x^{4}$

$3 x^{3}$

$3 x^{2}$

$10 x$

$15$

$x^{5}$

$0$

$5 x^{3}$

$2 x^{3}$

$3 x^{2}$

$10 x$

$2 x^{3}$

$0$

$10 x$

$3 x^{2}$

$0$

$15$

Step 12

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

						$x^{3}$	+	$0 x^{2}$	+	$2 x$	-	$3$
$x^{2}$	+	$0 x$	-	$5$		$x^{5}$	+	$0 x^{4}$	-	$3 x^{3}$	-	$3 x^{2}$	-	$10 x$	+	$15$
					-	$x^{5}$	-	$0$	+	$5 x^{3}$
									+	$2 x^{3}$	-	$3 x^{2}$	-	$10 x$
									-	$2 x^{3}$	-	$0$	+	$10 x$
											-	$3 x^{2}$	+	$0$	+	$15$

Step 13

Multiply the new quotient term by the divisor.

						$x^{3}$	+	$0 x^{2}$	+	$2 x$	-	$3$
$x^{2}$	+	$0 x$	-	$5$		$x^{5}$	+	$0 x^{4}$	-	$3 x^{3}$	-	$3 x^{2}$	-	$10 x$	+	$15$
					-	$x^{5}$	-	$0$	+	$5 x^{3}$
									+	$2 x^{3}$	-	$3 x^{2}$	-	$10 x$
									-	$2 x^{3}$	-	$0$	+	$10 x$
											-	$3 x^{2}$	+	$0$	+	$15$
											-	$3 x^{2}$	+	$0$	+	$15$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{2} + 0 + 15$

						$x^{3}$	+	$0 x^{2}$	+	$2 x$	-	$3$
$x^{2}$	+	$0 x$	-	$5$		$x^{5}$	+	$0 x^{4}$	-	$3 x^{3}$	-	$3 x^{2}$	-	$10 x$	+	$15$
					-	$x^{5}$	-	$0$	+	$5 x^{3}$
									+	$2 x^{3}$	-	$3 x^{2}$	-	$10 x$
									-	$2 x^{3}$	-	$0$	+	$10 x$
											-	$3 x^{2}$	+	$0$	+	$15$
											+	$3 x^{2}$	-	$0$	-	$15$

Step 15

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

						$x^{3}$	+	$0 x^{2}$	+	$2 x$	-	$3$
$x^{2}$	+	$0 x$	-	$5$		$x^{5}$	+	$0 x^{4}$	-	$3 x^{3}$	-	$3 x^{2}$	-	$10 x$	+	$15$
					-	$x^{5}$	-	$0$	+	$5 x^{3}$
									+	$2 x^{3}$	-	$3 x^{2}$	-	$10 x$
									-	$2 x^{3}$	-	$0$	+	$10 x$
											-	$3 x^{2}$	+	$0$	+	$15$
											+	$3 x^{2}$	-	$0$	-	$15$
																$0$

Step 16

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

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