Algebra Examples

$(x^{4} - 6 x^{2} - 23) \div (x + 2)$

Step 1

To calculate the remainder, first divide the polynomials.

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Step 1.1

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

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

Step 1.2

Divide the highest order term in the dividend $x^{4}$ by the highest order term in divisor $x$ .

$x^{3}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

Step 1.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{4} + 2 x^{3}$

$x^{3}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

Step 1.5

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

$x^{3}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

Step 1.6

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

$x^{3}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

Step 1.7

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

$x^{3}$

$2 x^{2}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

Step 1.8

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$2 x^{2}$
$x$	+	$2$		$x^{4}$	+	$0 x^{3}$	-	$6 x^{2}$	+	$0 x$	-	$23$
			-	$x^{4}$	-	$2 x^{3}$
					-	$2 x^{3}$	-	$6 x^{2}$
					-	$2 x^{3}$	-	$4 x^{2}$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{3} - 4 x^{2}$

$x^{3}$

$2 x^{2}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$2 x^{3}$

$4 x^{2}$

Step 1.10

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

$x^{3}$

$2 x^{2}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x^{2}$

Step 1.11

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

$x^{3}$

$2 x^{2}$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x^{2}$

$0 x$

Step 1.12

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

$x^{3}$

$2 x^{2}$

$2 x$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x^{2}$

$0 x$

Step 1.13

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x^{2}$

$2 x$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x^{2}$

$0 x$

$2 x^{2}$

$4 x$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{2} - 4 x$

$x^{3}$

$2 x^{2}$

$2 x$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x^{2}$

$0 x$

$2 x^{2}$

$4 x$

Step 1.15

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

$x^{3}$

$2 x^{2}$

$2 x$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x^{2}$

$0 x$

$2 x^{2}$

$4 x$

Step 1.16

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

$x^{3}$

$2 x^{2}$

$2 x$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x^{2}$

$0 x$

$2 x^{2}$

$4 x$

$23$

Step 1.17

Divide the highest order term in the dividend $4 x$ by the highest order term in divisor $x$ .

$x^{3}$

$2 x^{2}$

$2 x$

$4$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x^{2}$

$0 x$

$2 x^{2}$

$4 x$

$23$

Step 1.18

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x^{2}$

$2 x$

$4$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x^{2}$

$0 x$

$2 x^{2}$

$4 x$

$23$

$4 x$

$8$

Step 1.19

The expression needs to be subtracted from the dividend, so change all the signs in $4 x + 8$

$x^{3}$

$2 x^{2}$

$2 x$

$4$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x^{2}$

$0 x$

$2 x^{2}$

$4 x$

$23$

$4 x$

$8$

Step 1.20

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

$x^{3}$

$2 x^{2}$

$2 x$

$4$

$x$

$2$

$x^{4}$

$0 x^{3}$

$6 x^{2}$

$0 x$

$23$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$2 x^{3}$

$4 x^{2}$

$2 x^{2}$

$0 x$

$2 x^{2}$

$4 x$

$23$

$4 x$

$8$

$31$

Step 1.21

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

$x^{3} - 2 x^{2} - 2 x + 4 - \frac{31}{x + 2}$

Step 2

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

$- 31$