Algebra Examples

x^{3} - 5 x + 6

$x^{3} - 5 x + 6$ ,

x + 2

$x + 2$

Step 1

Divide the higher order polynomial by the other polynomial in order to find the remainder.

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

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

Step 2

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

0

$0$ .

x

$x$

2

$2$

x^{3}

$x^{3}$

0 x^{2}

$0 x^{2}$

5 x

$5 x$

6

$6$

Step 3

Divide the highest order term in the dividend

x^{3}

$x^{3}$ by the highest order term in divisor

x

$x$ .

					$x^{2}$ $x^{2}$
	$x$ $x$	+	$2$ $2$		$x^{3}$ $x^{3}$	+	$0 x^{2}$ $0 x^{2}$	-	$5 x$ $5 x$	+	$6$ $6$

Step 4

Multiply the new quotient term by the divisor.

				$x^{2}$ $x^{2}$
$x$ $x$	+	$2$ $2$		$x^{3}$ $x^{3}$	+	$0 x^{2}$ $0 x^{2}$	-	$5 x$ $5 x$	+	$6$ $6$
			+	$x^{3}$ $x^{3}$	+	$2 x^{2}$ $2 x^{2}$

Step 5

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

x^{3} + 2 x^{2}

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

				$x^{2}$ $x^{2}$
$x$ $x$	+	$2$ $2$		$x^{3}$ $x^{3}$	+	$0 x^{2}$ $0 x^{2}$	-	$5 x$ $5 x$	+	$6$ $6$
			-	$x^{3}$ $x^{3}$	-	$2 x^{2}$ $2 x^{2}$

Step 6

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

				$x^{2}$ $x^{2}$
$x$ $x$	+	$2$ $2$		$x^{3}$ $x^{3}$	+	$0 x^{2}$ $0 x^{2}$	-	$5 x$ $5 x$	+	$6$ $6$
			-	$x^{3}$ $x^{3}$	-	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$

Step 7

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

				$x^{2}$ $x^{2}$
$x$ $x$	+	$2$ $2$		$x^{3}$ $x^{3}$	+	$0 x^{2}$ $0 x^{2}$	-	$5 x$ $5 x$	+	$6$ $6$
			-	$x^{3}$ $x^{3}$	-	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$5 x$ $5 x$

Step 8

Divide the highest order term in the dividend

- 2 x^{2}

$- 2 x^{2}$ by the highest order term in divisor

x

$x$ .

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$
$x$ $x$	+	$2$ $2$		$x^{3}$ $x^{3}$	+	$0 x^{2}$ $0 x^{2}$	-	$5 x$ $5 x$	+	$6$ $6$
			-	$x^{3}$ $x^{3}$	-	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$5 x$ $5 x$

Step 9

Multiply the new quotient term by the divisor.

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$
$x$ $x$	+	$2$ $2$		$x^{3}$ $x^{3}$	+	$0 x^{2}$ $0 x^{2}$	-	$5 x$ $5 x$	+	$6$ $6$
			-	$x^{3}$ $x^{3}$	-	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$5 x$ $5 x$
					-	$2 x^{2}$ $2 x^{2}$	-	$4 x$ $4 x$

Step 10

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

- 2 x^{2} - 4 x

$- 2 x^{2} - 4 x$

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$
$x$ $x$	+	$2$ $2$		$x^{3}$ $x^{3}$	+	$0 x^{2}$ $0 x^{2}$	-	$5 x$ $5 x$	+	$6$ $6$
			-	$x^{3}$ $x^{3}$	-	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$5 x$ $5 x$
					+	$2 x^{2}$ $2 x^{2}$	+	$4 x$ $4 x$

Step 11

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

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$
$x$ $x$	+	$2$ $2$		$x^{3}$ $x^{3}$	+	$0 x^{2}$ $0 x^{2}$	-	$5 x$ $5 x$	+	$6$ $6$
			-	$x^{3}$ $x^{3}$	-	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$5 x$ $5 x$
					+	$2 x^{2}$ $2 x^{2}$	+	$4 x$ $4 x$
							-	$x$ $x$

Step 12

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

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$
$x$ $x$	+	$2$ $2$		$x^{3}$ $x^{3}$	+	$0 x^{2}$ $0 x^{2}$	-	$5 x$ $5 x$	+	$6$ $6$
			-	$x^{3}$ $x^{3}$	-	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$5 x$ $5 x$
					+	$2 x^{2}$ $2 x^{2}$	+	$4 x$ $4 x$
							-	$x$ $x$	+	$6$ $6$

Step 13

Divide the highest order term in the dividend

- x

$- x$ by the highest order term in divisor

x

$x$ .

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$	-	$1$ $1$
$x$ $x$	+	$2$ $2$		$x^{3}$ $x^{3}$	+	$0 x^{2}$ $0 x^{2}$	-	$5 x$ $5 x$	+	$6$ $6$
			-	$x^{3}$ $x^{3}$	-	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$5 x$ $5 x$
					+	$2 x^{2}$ $2 x^{2}$	+	$4 x$ $4 x$
							-	$x$ $x$	+	$6$ $6$

Step 14

Multiply the new quotient term by the divisor.

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$	-	$1$ $1$
$x$ $x$	+	$2$ $2$		$x^{3}$ $x^{3}$	+	$0 x^{2}$ $0 x^{2}$	-	$5 x$ $5 x$	+	$6$ $6$
			-	$x^{3}$ $x^{3}$	-	$2 x^{2}$ $2 x^{2}$
					-	$2 x^{2}$ $2 x^{2}$	-	$5 x$ $5 x$
					+	$2 x^{2}$ $2 x^{2}$	+	$4 x$ $4 x$
							-	$x$ $x$	+	$6$ $6$
							-	$x$ $x$	-	$2$ $2$

Step 15

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

- x - 2

$- x - 2$

				$x^{2}$ $x^{2}$	-	$2 x$ $2 x$	-	$1$ $1$
$x$ $x$	+	$2$ $2$		$x^{3}$ $x^{3}$	+	$0 x^{2}$ $0 x^{2}$	-	$5 x$ $5 x$	+	$6$ $6$
			-	$x^{3}$ $x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$4 x$
							-	$x$	+	$6$
							+	$x$	+	$2$

Step 16

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

				$x^{2}$	-	$2 x$	-	$1$
$x$	+	$2$		$x^{3}$	+	$0 x^{2}$	-	$5 x$	+	$6$
			-	$x^{3}$	-	$2 x^{2}$
					-	$2 x^{2}$	-	$5 x$
					+	$2 x^{2}$	+	$4 x$
							-	$x$	+	$6$
							+	$x$	+	$2$
									+	$8$

Step 17

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

$x^{2} - 2 x - 1 + \frac{8}{x + 2}$

Step 18

The remainder is the part of the answer that is left after the division by

$x + 2$ is complete.

$8$

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