Algebra Examples

$\frac{4 x^{5} + 5 x^{4} + 1}{x - 2}$

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 $0$ .

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 1.2

Divide the highest order term in the dividend $4 x^{5}$ by the highest order term in divisor $x$ .

$4 x^{4}$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 1.3

Multiply the new quotient term by the divisor.

$4 x^{4}$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{5} - 8 x^{4}$

$4 x^{4}$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

Step 1.5

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

$4 x^{4}$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

Step 1.6

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

$4 x^{4}$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

Step 1.7

Divide the highest order term in the dividend $13 x^{4}$ by the highest order term in divisor $x$ .

$4 x^{4}$

$13 x^{3}$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

Step 1.8

Multiply the new quotient term by the divisor.

$4 x^{4}$

$13 x^{3}$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $13 x^{4} - 26 x^{3}$

$4 x^{4}$

$13 x^{3}$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

Step 1.10

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

$4 x^{4}$

$13 x^{3}$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

Step 1.11

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

$4 x^{4}$

$13 x^{3}$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

$0 x^{2}$

Step 1.12

Divide the highest order term in the dividend $26 x^{3}$ by the highest order term in divisor $x$ .

$4 x^{4}$

$13 x^{3}$

$26 x^{2}$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

$0 x^{2}$

Step 1.13

Multiply the new quotient term by the divisor.

$4 x^{4}$

$13 x^{3}$

$26 x^{2}$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

$0 x^{2}$

$26 x^{3}$

$52 x^{2}$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $26 x^{3} - 52 x^{2}$

$4 x^{4}$

$13 x^{3}$

$26 x^{2}$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

$0 x^{2}$

$26 x^{3}$

$52 x^{2}$

Step 1.15

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

$4 x^{4}$

$13 x^{3}$

$26 x^{2}$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

$0 x^{2}$

$26 x^{3}$

$52 x^{2}$

Step 1.16

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

$4 x^{4}$

$13 x^{3}$

$26 x^{2}$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

$0 x^{2}$

$26 x^{3}$

$52 x^{2}$

$0 x$

Step 1.17

Divide the highest order term in the dividend $52 x^{2}$ by the highest order term in divisor $x$ .

$4 x^{4}$

$13 x^{3}$

$26 x^{2}$

$52 x$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

$0 x^{2}$

$26 x^{3}$

$52 x^{2}$

$0 x$

Step 1.18

Multiply the new quotient term by the divisor.

$4 x^{4}$

$13 x^{3}$

$26 x^{2}$

$52 x$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

$0 x^{2}$

$26 x^{3}$

$52 x^{2}$

$0 x$

$52 x^{2}$

$104 x$

Step 1.19

The expression needs to be subtracted from the dividend, so change all the signs in $52 x^{2} - 104 x$

$4 x^{4}$

$13 x^{3}$

$26 x^{2}$

$52 x$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

$0 x^{2}$

$26 x^{3}$

$52 x^{2}$

$0 x$

$52 x^{2}$

$104 x$

Step 1.20

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

$4 x^{4}$

$13 x^{3}$

$26 x^{2}$

$52 x$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

$0 x^{2}$

$26 x^{3}$

$52 x^{2}$

$0 x$

$52 x^{2}$

$104 x$

Step 1.21

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

$4 x^{4}$

$13 x^{3}$

$26 x^{2}$

$52 x$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

$0 x^{2}$

$26 x^{3}$

$52 x^{2}$

$0 x$

$52 x^{2}$

$104 x$

$1$

Step 1.22

Divide the highest order term in the dividend $104 x$ by the highest order term in divisor $x$ .

$4 x^{4}$

$13 x^{3}$

$26 x^{2}$

$52 x$

$104$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

$0 x^{2}$

$26 x^{3}$

$52 x^{2}$

$0 x$

$52 x^{2}$

$104 x$

$1$

Step 1.23

Multiply the new quotient term by the divisor.

$4 x^{4}$

$13 x^{3}$

$26 x^{2}$

$52 x$

$104$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

$0 x^{2}$

$26 x^{3}$

$52 x^{2}$

$0 x$

$52 x^{2}$

$104 x$

$1$

$104 x$

$208$

Step 1.24

The expression needs to be subtracted from the dividend, so change all the signs in $104 x - 208$

$4 x^{4}$

$13 x^{3}$

$26 x^{2}$

$52 x$

$104$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

$0 x^{2}$

$26 x^{3}$

$52 x^{2}$

$0 x$

$52 x^{2}$

$104 x$

$1$

$104 x$

$208$

Step 1.25

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

$4 x^{4}$

$13 x^{3}$

$26 x^{2}$

$52 x$

$104$

$x$

$2$

$4 x^{5}$

$5 x^{4}$

$0 x^{3}$

$0 x^{2}$

$0 x$

$1$

$4 x^{5}$

$8 x^{4}$

$13 x^{4}$

$0 x^{3}$

$13 x^{4}$

$26 x^{3}$

$0 x^{2}$

$26 x^{3}$

$52 x^{2}$

$0 x$

$52 x^{2}$

$104 x$

$1$

$104 x$

$208$

$209$

Step 1.26

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

$4 x^{4} + 13 x^{3} + 26 x^{2} + 52 x + 104 + \frac{209}{x - 2}$

Step 2

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

$209$