Algebra Examples

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

Step 1

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

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

Step 2

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

$x^{4}$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$x^{4}$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{5}$

$0$

Step 4

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

$x^{4}$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{5}$

$0$

Step 5

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

$x^{4}$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{5}$

$0$

Step 6

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

$x^{4}$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{5}$

$0$

$0 x^{3}$

$3 x^{2}$

Step 7

Divide the highest order term in the dividend $0 x^{3}$ by the highest order term in divisor $x$ .

$x^{4}$

$0 x^{3}$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{5}$

$0$

$0 x^{3}$

$3 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$x^{4}$

$0 x^{3}$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{5}$

$0$

$0 x^{3}$

$3 x^{2}$

$0$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $0 + 0$

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

Step 10

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

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

Step 11

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

$x^{4}$

$0 x^{3}$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{5}$

$0$

$0 x^{3}$

$3 x^{2}$

$0$

$3 x^{2}$

$2 x$

Step 12

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

$x^{4}$

$0 x^{3}$

$3 x$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{5}$

$0$

$0 x^{3}$

$3 x^{2}$

$0$

$3 x^{2}$

$2 x$

Step 13

Multiply the new quotient term by the divisor.

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

Step 14

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

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

Step 15

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

$x^{4}$

$0 x^{3}$

$3 x$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{5}$

$0$

$0 x^{3}$

$3 x^{2}$

$0$

$3 x^{2}$

$2 x$

$3 x^{2}$

$0$

$2 x$

Step 16

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

$x^{4}$

$0 x^{3}$

$3 x$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{5}$

$0$

$0 x^{3}$

$3 x^{2}$

$0$

$3 x^{2}$

$2 x$

$3 x^{2}$

$0$

$2 x$

$0$

Step 17

Divide the highest order term in the dividend $2 x$ by the highest order term in divisor $x$ .

$x^{4}$

$0 x^{3}$

$3 x$

$2$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{5}$

$0$

$0 x^{3}$

$3 x^{2}$

$0$

$3 x^{2}$

$2 x$

$3 x^{2}$

$0$

$2 x$

$0$

Step 18

Multiply the new quotient term by the divisor.

$x^{4}$

$0 x^{3}$

$3 x$

$2$

$x$

$0$

$x^{5}$

$0 x^{4}$

$0 x^{3}$

$3 x^{2}$

$2 x$

$0$

$x^{5}$

$0$

$0 x^{3}$

$3 x^{2}$

$0$

$3 x^{2}$

$2 x$

$3 x^{2}$

$0$

$2 x$

$0$

$2 x$

$0$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $2 x + 0$

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

Step 20

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

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

Step 21

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

$x^{4} + 0 x^{3} - 3 x + 2$