Algebra Examples

$\frac{(x^{5} + x^{3} - 1)}{(x - 4)}$

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 $0$ .

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 1.2

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

$x^{4}$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

Step 1.3

Multiply the new quotient term by the divisor.

$x^{4}$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

Step 1.4

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

$x^{4}$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

Step 1.5

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

$x^{4}$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

Step 1.6

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

$x^{4}$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

Step 1.7

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

$x^{4}$

$4 x^{3}$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

Step 1.8

Multiply the new quotient term by the divisor.

$x^{4}$

$4 x^{3}$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{4} - 16 x^{3}$

$x^{4}$

$4 x^{3}$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

Step 1.10

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

$x^{4}$

$4 x^{3}$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

$17 x^{3}$

Step 1.11

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

$x^{4}$

$4 x^{3}$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

$17 x^{3}$

$0 x^{2}$

Step 1.12

Divide the highest order term in the dividend $17 x^{3}$ by the highest order term in divisor $x$ .

$x^{4}$

$4 x^{3}$

$17 x^{2}$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

$17 x^{3}$

$0 x^{2}$

Step 1.13

Multiply the new quotient term by the divisor.

$x^{4}$

$4 x^{3}$

$17 x^{2}$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

$17 x^{3}$

$0 x^{2}$

$17 x^{3}$

$68 x^{2}$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $17 x^{3} - 68 x^{2}$

$x^{4}$

$4 x^{3}$

$17 x^{2}$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

$17 x^{3}$

$0 x^{2}$

$17 x^{3}$

$68 x^{2}$

Step 1.15

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

$x^{4}$

$4 x^{3}$

$17 x^{2}$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

$17 x^{3}$

$0 x^{2}$

$17 x^{3}$

$68 x^{2}$

Step 1.16

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

$x^{4}$

$4 x^{3}$

$17 x^{2}$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

$17 x^{3}$

$0 x^{2}$

$17 x^{3}$

$68 x^{2}$

$0 x$

Step 1.17

Divide the highest order term in the dividend $68 x^{2}$ by the highest order term in divisor $x$ .

$x^{4}$

$4 x^{3}$

$17 x^{2}$

$68 x$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

$17 x^{3}$

$0 x^{2}$

$17 x^{3}$

$68 x^{2}$

$0 x$

Step 1.18

Multiply the new quotient term by the divisor.

$x^{4}$

$4 x^{3}$

$17 x^{2}$

$68 x$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

$17 x^{3}$

$0 x^{2}$

$17 x^{3}$

$68 x^{2}$

$0 x$

$68 x^{2}$

$272 x$

Step 1.19

The expression needs to be subtracted from the dividend, so change all the signs in $68 x^{2} - 272 x$

$x^{4}$

$4 x^{3}$

$17 x^{2}$

$68 x$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

$17 x^{3}$

$0 x^{2}$

$17 x^{3}$

$68 x^{2}$

$0 x$

$68 x^{2}$

$272 x$

Step 1.20

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

$x^{4}$

$4 x^{3}$

$17 x^{2}$

$68 x$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

$17 x^{3}$

$0 x^{2}$

$17 x^{3}$

$68 x^{2}$

$0 x$

$68 x^{2}$

$272 x$

Step 1.21

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

$x^{4}$

$4 x^{3}$

$17 x^{2}$

$68 x$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

$17 x^{3}$

$0 x^{2}$

$17 x^{3}$

$68 x^{2}$

$0 x$

$68 x^{2}$

$272 x$

$1$

Step 1.22

Divide the highest order term in the dividend $272 x$ by the highest order term in divisor $x$ .

$x^{4}$

$4 x^{3}$

$17 x^{2}$

$68 x$

$272$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

$17 x^{3}$

$0 x^{2}$

$17 x^{3}$

$68 x^{2}$

$0 x$

$68 x^{2}$

$272 x$

$1$

Step 1.23

Multiply the new quotient term by the divisor.

$x^{4}$

$4 x^{3}$

$17 x^{2}$

$68 x$

$272$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

$17 x^{3}$

$0 x^{2}$

$17 x^{3}$

$68 x^{2}$

$0 x$

$68 x^{2}$

$272 x$

$1$

$272 x$

$1088$

Step 1.24

The expression needs to be subtracted from the dividend, so change all the signs in $272 x - 1088$

$x^{4}$

$4 x^{3}$

$17 x^{2}$

$68 x$

$272$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

$17 x^{3}$

$0 x^{2}$

$17 x^{3}$

$68 x^{2}$

$0 x$

$68 x^{2}$

$272 x$

$1$

$272 x$

$1088$

Step 1.25

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

$x^{4}$

$4 x^{3}$

$17 x^{2}$

$68 x$

$272$

$x$

$4$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$0 x$

$1$

$x^{5}$

$4 x^{4}$

$x^{3}$

$4 x^{4}$

$16 x^{3}$

$17 x^{3}$

$0 x^{2}$

$17 x^{3}$

$68 x^{2}$

$0 x$

$68 x^{2}$

$272 x$

$1$

$272 x$

$1088$

$1087$

Step 1.26

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

$x^{4} + 4 x^{3} + 17 x^{2} + 68 x + 272 + \frac{1087}{x - 4}$

Step 2

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

$1087$