Algebra Examples

$(9 x^{4} - 45 x^{3} + 37 x^{2} + x + 2) \div (x - 2)$

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 $0$ .

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

Step 1.2

Divide the highest order term in the dividend $9 x^{4}$ by the highest order term in divisor $x$ .

$9 x^{3}$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

Step 1.3

Multiply the new quotient term by the divisor.

$9 x^{3}$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $9 x^{4} - 18 x^{3}$

$9 x^{3}$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

Step 1.5

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

$9 x^{3}$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

$27 x^{3}$

Step 1.6

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

$9 x^{3}$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

$27 x^{3}$

$37 x^{2}$

Step 1.7

Divide the highest order term in the dividend $- 27 x^{3}$ by the highest order term in divisor $x$ .

$9 x^{3}$

$27 x^{2}$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

$27 x^{3}$

$37 x^{2}$

Step 1.8

Multiply the new quotient term by the divisor.

$9 x^{3}$

$27 x^{2}$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

$27 x^{3}$

$37 x^{2}$

$27 x^{3}$

$54 x^{2}$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 27 x^{3} + 54 x^{2}$

$9 x^{3}$

$27 x^{2}$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

$27 x^{3}$

$37 x^{2}$

$27 x^{3}$

$54 x^{2}$

Step 1.10

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

$9 x^{3}$

$27 x^{2}$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

$27 x^{3}$

$37 x^{2}$

$27 x^{3}$

$54 x^{2}$

$17 x^{2}$

Step 1.11

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

$9 x^{3}$

$27 x^{2}$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

$27 x^{3}$

$37 x^{2}$

$27 x^{3}$

$54 x^{2}$

$17 x^{2}$

$x$

Step 1.12

Divide the highest order term in the dividend $- 17 x^{2}$ by the highest order term in divisor $x$ .

$9 x^{3}$

$27 x^{2}$

$17 x$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

$27 x^{3}$

$37 x^{2}$

$27 x^{3}$

$54 x^{2}$

$17 x^{2}$

$x$

Step 1.13

Multiply the new quotient term by the divisor.

$9 x^{3}$

$27 x^{2}$

$17 x$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

$27 x^{3}$

$37 x^{2}$

$27 x^{3}$

$54 x^{2}$

$17 x^{2}$

$x$

$17 x^{2}$

$34 x$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 17 x^{2} + 34 x$

$9 x^{3}$

$27 x^{2}$

$17 x$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

$27 x^{3}$

$37 x^{2}$

$27 x^{3}$

$54 x^{2}$

$17 x^{2}$

$x$

$17 x^{2}$

$34 x$

Step 1.15

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

$9 x^{3}$

$27 x^{2}$

$17 x$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

$27 x^{3}$

$37 x^{2}$

$27 x^{3}$

$54 x^{2}$

$17 x^{2}$

$x$

$17 x^{2}$

$34 x$

$33 x$

Step 1.16

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

$9 x^{3}$

$27 x^{2}$

$17 x$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

$27 x^{3}$

$37 x^{2}$

$27 x^{3}$

$54 x^{2}$

$17 x^{2}$

$x$

$17 x^{2}$

$34 x$

$33 x$

$2$

Step 1.17

Divide the highest order term in the dividend $- 33 x$ by the highest order term in divisor $x$ .

$9 x^{3}$

$27 x^{2}$

$17 x$

$33$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

$27 x^{3}$

$37 x^{2}$

$27 x^{3}$

$54 x^{2}$

$17 x^{2}$

$x$

$17 x^{2}$

$34 x$

$33 x$

$2$

Step 1.18

Multiply the new quotient term by the divisor.

$9 x^{3}$

$27 x^{2}$

$17 x$

$33$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

$27 x^{3}$

$37 x^{2}$

$27 x^{3}$

$54 x^{2}$

$17 x^{2}$

$x$

$17 x^{2}$

$34 x$

$33 x$

$2$

$33 x$

$66$

Step 1.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 33 x + 66$

$9 x^{3}$

$27 x^{2}$

$17 x$

$33$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

$27 x^{3}$

$37 x^{2}$

$27 x^{3}$

$54 x^{2}$

$17 x^{2}$

$x$

$17 x^{2}$

$34 x$

$33 x$

$2$

$33 x$

$66$

Step 1.20

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

$9 x^{3}$

$27 x^{2}$

$17 x$

$33$

$x$

$2$

$9 x^{4}$

$45 x^{3}$

$37 x^{2}$

$x$

$2$

$9 x^{4}$

$18 x^{3}$

$27 x^{3}$

$37 x^{2}$

$27 x^{3}$

$54 x^{2}$

$17 x^{2}$

$x$

$17 x^{2}$

$34 x$

$33 x$

$2$

$33 x$

$66$

$64$

Step 1.21

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

$9 x^{3} - 27 x^{2} - 17 x - 33 - \frac{64}{x - 2}$

Step 2

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

$- 64$