Algebra Examples

$\frac{25 x^{5} - 5 x^{4} + 14 x^{3} + 8 x^{2} + 4}{5 x^{2} + 2 x - 1}$

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 $0$ .

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

Step 1.2

Divide the highest order term in the dividend $25 x^{5}$ by the highest order term in divisor $5 x^{2}$ .

$5 x^{3}$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

Step 1.3

Multiply the new quotient term by the divisor.

$5 x^{3}$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

Step 1.4

The expression needs to be subtracted from the dividend, so change all the signs in $25 x^{5} + 10 x^{4} - 5 x^{3}$

$5 x^{3}$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

Step 1.5

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

$5 x^{3}$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

$15 x^{4}$

$19 x^{3}$

Step 1.6

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

$5 x^{3}$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

$15 x^{4}$

$19 x^{3}$

$8 x^{2}$

Step 1.7

Divide the highest order term in the dividend $- 15 x^{4}$ by the highest order term in divisor $5 x^{2}$ .

$5 x^{3}$

$3 x^{2}$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

$15 x^{4}$

$19 x^{3}$

$8 x^{2}$

Step 1.8

Multiply the new quotient term by the divisor.

$5 x^{3}$

$3 x^{2}$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

$15 x^{4}$

$19 x^{3}$

$8 x^{2}$

$15 x^{4}$

$6 x^{3}$

$3 x^{2}$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x^{4} - 6 x^{3} + 3 x^{2}$

$5 x^{3}$

$3 x^{2}$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

$15 x^{4}$

$19 x^{3}$

$8 x^{2}$

$15 x^{4}$

$6 x^{3}$

$3 x^{2}$

Step 1.10

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

$5 x^{3}$

$3 x^{2}$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

$15 x^{4}$

$19 x^{3}$

$8 x^{2}$

$15 x^{4}$

$6 x^{3}$

$3 x^{2}$

$25 x^{3}$

$5 x^{2}$

Step 1.11

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

$5 x^{3}$

$3 x^{2}$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

$15 x^{4}$

$19 x^{3}$

$8 x^{2}$

$15 x^{4}$

$6 x^{3}$

$3 x^{2}$

$25 x^{3}$

$5 x^{2}$

$0 x$

Step 1.12

Divide the highest order term in the dividend $25 x^{3}$ by the highest order term in divisor $5 x^{2}$ .

$5 x^{3}$

$3 x^{2}$

$5 x$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

$15 x^{4}$

$19 x^{3}$

$8 x^{2}$

$15 x^{4}$

$6 x^{3}$

$3 x^{2}$

$25 x^{3}$

$5 x^{2}$

$0 x$

Step 1.13

Multiply the new quotient term by the divisor.

$5 x^{3}$

$3 x^{2}$

$5 x$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

$15 x^{4}$

$19 x^{3}$

$8 x^{2}$

$15 x^{4}$

$6 x^{3}$

$3 x^{2}$

$25 x^{3}$

$5 x^{2}$

$0 x$

$25 x^{3}$

$10 x^{2}$

$5 x$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $25 x^{3} + 10 x^{2} - 5 x$

$5 x^{3}$

$3 x^{2}$

$5 x$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

$15 x^{4}$

$19 x^{3}$

$8 x^{2}$

$15 x^{4}$

$6 x^{3}$

$3 x^{2}$

$25 x^{3}$

$5 x^{2}$

$0 x$

$25 x^{3}$

$10 x^{2}$

$5 x$

Step 1.15

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

$5 x^{3}$

$3 x^{2}$

$5 x$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

$15 x^{4}$

$19 x^{3}$

$8 x^{2}$

$15 x^{4}$

$6 x^{3}$

$3 x^{2}$

$25 x^{3}$

$5 x^{2}$

$0 x$

$25 x^{3}$

$10 x^{2}$

$5 x$

$5 x^{2}$

$5 x$

Step 1.16

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

$5 x^{3}$

$3 x^{2}$

$5 x$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

$15 x^{4}$

$19 x^{3}$

$8 x^{2}$

$15 x^{4}$

$6 x^{3}$

$3 x^{2}$

$25 x^{3}$

$5 x^{2}$

$0 x$

$25 x^{3}$

$10 x^{2}$

$5 x$

$5 x^{2}$

$5 x$

$4$

Step 1.17

Divide the highest order term in the dividend $- 5 x^{2}$ by the highest order term in divisor $5 x^{2}$ .

$5 x^{3}$

$3 x^{2}$

$5 x$

$1$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

$15 x^{4}$

$19 x^{3}$

$8 x^{2}$

$15 x^{4}$

$6 x^{3}$

$3 x^{2}$

$25 x^{3}$

$5 x^{2}$

$0 x$

$25 x^{3}$

$10 x^{2}$

$5 x$

$5 x^{2}$

$5 x$

$4$

Step 1.18

Multiply the new quotient term by the divisor.

$5 x^{3}$

$3 x^{2}$

$5 x$

$1$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

$15 x^{4}$

$19 x^{3}$

$8 x^{2}$

$15 x^{4}$

$6 x^{3}$

$3 x^{2}$

$25 x^{3}$

$5 x^{2}$

$0 x$

$25 x^{3}$

$10 x^{2}$

$5 x$

$5 x^{2}$

$5 x$

$4$

$5 x^{2}$

$2 x$

$1$

Step 1.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{2} - 2 x + 1$

$5 x^{3}$

$3 x^{2}$

$5 x$

$1$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

$15 x^{4}$

$19 x^{3}$

$8 x^{2}$

$15 x^{4}$

$6 x^{3}$

$3 x^{2}$

$25 x^{3}$

$5 x^{2}$

$0 x$

$25 x^{3}$

$10 x^{2}$

$5 x$

$5 x^{2}$

$5 x$

$4$

$5 x^{2}$

$2 x$

$1$

Step 1.20

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

$5 x^{3}$

$3 x^{2}$

$5 x$

$1$

$5 x^{2}$

$2 x$

$1$

$25 x^{5}$

$5 x^{4}$

$14 x^{3}$

$8 x^{2}$

$0 x$

$4$

$25 x^{5}$

$10 x^{4}$

$5 x^{3}$

$15 x^{4}$

$19 x^{3}$

$8 x^{2}$

$15 x^{4}$

$6 x^{3}$

$3 x^{2}$

$25 x^{3}$

$5 x^{2}$

$0 x$

$25 x^{3}$

$10 x^{2}$

$5 x$

$5 x^{2}$

$5 x$

$4$

$5 x^{2}$

$2 x$

$1$

$7 x$

$3$

Step 1.21

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

$5 x^{3} - 3 x^{2} + 5 x - 1 + \frac{7 x + 3}{5 x^{2} + 2 x - 1}$

Step 2

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

$7 x + 3$