Algebra Examples

$7 x^{3} - 8 x^{2} + x + 1$ divided by $x + 2$

Step 1

Write the problem as a mathematical expression.

$\frac{7 x^{3} - 8 x^{2} + x + 1}{x + 2}$

Step 2

To calculate the remainder, first divide the polynomials.

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Step 2.1

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

$x$

$2$

$7 x^{3}$

$8 x^{2}$

$x$

$1$

Step 2.2

Divide the highest order term in the dividend $7 x^{3}$ by the highest order term in divisor $x$ .

					$7 x^{2}$
	$x$	+	$2$		$7 x^{3}$	-	$8 x^{2}$	+	$x$	+	$1$

Step 2.3

Multiply the new quotient term by the divisor.

				$7 x^{2}$
$x$	+	$2$		$7 x^{3}$	-	$8 x^{2}$	+	$x$	+	$1$
			+	$7 x^{3}$	+	$14 x^{2}$

Step 2.4

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{3} + 14 x^{2}$

				$7 x^{2}$
$x$	+	$2$		$7 x^{3}$	-	$8 x^{2}$	+	$x$	+	$1$
			-	$7 x^{3}$	-	$14 x^{2}$

Step 2.5

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

				$7 x^{2}$
$x$	+	$2$		$7 x^{3}$	-	$8 x^{2}$	+	$x$	+	$1$
			-	$7 x^{3}$	-	$14 x^{2}$
					-	$22 x^{2}$

Step 2.6

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

				$7 x^{2}$
$x$	+	$2$		$7 x^{3}$	-	$8 x^{2}$	+	$x$	+	$1$
			-	$7 x^{3}$	-	$14 x^{2}$
					-	$22 x^{2}$	+	$x$

Step 2.7

Divide the highest order term in the dividend $- 22 x^{2}$ by the highest order term in divisor $x$ .

				$7 x^{2}$	-	$22 x$
$x$	+	$2$		$7 x^{3}$	-	$8 x^{2}$	+	$x$	+	$1$
			-	$7 x^{3}$	-	$14 x^{2}$
					-	$22 x^{2}$	+	$x$

Step 2.8

Multiply the new quotient term by the divisor.

				$7 x^{2}$	-	$22 x$
$x$	+	$2$		$7 x^{3}$	-	$8 x^{2}$	+	$x$	+	$1$
			-	$7 x^{3}$	-	$14 x^{2}$
					-	$22 x^{2}$	+	$x$
					-	$22 x^{2}$	-	$44 x$

Step 2.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 22 x^{2} - 44 x$

				$7 x^{2}$	-	$22 x$
$x$	+	$2$		$7 x^{3}$	-	$8 x^{2}$	+	$x$	+	$1$
			-	$7 x^{3}$	-	$14 x^{2}$
					-	$22 x^{2}$	+	$x$
					+	$22 x^{2}$	+	$44 x$

Step 2.10

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

				$7 x^{2}$	-	$22 x$
$x$	+	$2$		$7 x^{3}$	-	$8 x^{2}$	+	$x$	+	$1$
			-	$7 x^{3}$	-	$14 x^{2}$
					-	$22 x^{2}$	+	$x$
					+	$22 x^{2}$	+	$44 x$
							+	$45 x$

Step 2.11

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

				$7 x^{2}$	-	$22 x$
$x$	+	$2$		$7 x^{3}$	-	$8 x^{2}$	+	$x$	+	$1$
			-	$7 x^{3}$	-	$14 x^{2}$
					-	$22 x^{2}$	+	$x$
					+	$22 x^{2}$	+	$44 x$
							+	$45 x$	+	$1$

Step 2.12

Divide the highest order term in the dividend $45 x$ by the highest order term in divisor $x$ .

				$7 x^{2}$	-	$22 x$	+	$45$
$x$	+	$2$		$7 x^{3}$	-	$8 x^{2}$	+	$x$	+	$1$
			-	$7 x^{3}$	-	$14 x^{2}$
					-	$22 x^{2}$	+	$x$
					+	$22 x^{2}$	+	$44 x$
							+	$45 x$	+	$1$

Step 2.13

Multiply the new quotient term by the divisor.

				$7 x^{2}$	-	$22 x$	+	$45$
$x$	+	$2$		$7 x^{3}$	-	$8 x^{2}$	+	$x$	+	$1$
			-	$7 x^{3}$	-	$14 x^{2}$
					-	$22 x^{2}$	+	$x$
					+	$22 x^{2}$	+	$44 x$
							+	$45 x$	+	$1$
							+	$45 x$	+	$90$

Step 2.14

The expression needs to be subtracted from the dividend, so change all the signs in $45 x + 90$

				$7 x^{2}$	-	$22 x$	+	$45$
$x$	+	$2$		$7 x^{3}$	-	$8 x^{2}$	+	$x$	+	$1$
			-	$7 x^{3}$	-	$14 x^{2}$
					-	$22 x^{2}$	+	$x$
					+	$22 x^{2}$	+	$44 x$
							+	$45 x$	+	$1$
							-	$45 x$	-	$90$

Step 2.15

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

				$7 x^{2}$	-	$22 x$	+	$45$
$x$	+	$2$		$7 x^{3}$	-	$8 x^{2}$	+	$x$	+	$1$
			-	$7 x^{3}$	-	$14 x^{2}$
					-	$22 x^{2}$	+	$x$
					+	$22 x^{2}$	+	$44 x$
							+	$45 x$	+	$1$
							-	$45 x$	-	$90$
									-	$89$

Step 2.16

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

$7 x^{2} - 22 x + 45 - \frac{89}{x + 2}$

Step 3

Since the last term in the resulting expression is a fraction, the numerator of the fraction is the remainder.

$- 89$