Algebra Examples

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

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 $0$ .

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

Step 1.2

Divide the highest order term in the dividend $3 x^{4}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

Step 1.3

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

Step 1.4

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

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

Step 1.5

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

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

$8 x^{3}$

Step 1.6

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

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

$8 x^{3}$

$3 x^{2}$

Step 1.7

Divide the highest order term in the dividend $- 8 x^{3}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$8 x^{2}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

$8 x^{3}$

$3 x^{2}$

Step 1.8

Multiply the new quotient term by the divisor.

$3 x^{3}$

$8 x^{2}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

$8 x^{3}$

$3 x^{2}$

$8 x^{3}$

$16 x^{2}$

Step 1.9

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{3} - 16 x^{2}$

$3 x^{3}$

$8 x^{2}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

$8 x^{3}$

$3 x^{2}$

$8 x^{3}$

$16 x^{2}$

Step 1.10

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

$3 x^{3}$

$8 x^{2}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

$8 x^{3}$

$3 x^{2}$

$8 x^{3}$

$16 x^{2}$

$19 x^{2}$

Step 1.11

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

$3 x^{3}$

$8 x^{2}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

$8 x^{3}$

$3 x^{2}$

$8 x^{3}$

$16 x^{2}$

$19 x^{2}$

$2 x$

Step 1.12

Divide the highest order term in the dividend $19 x^{2}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$8 x^{2}$

$19 x$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

$8 x^{3}$

$3 x^{2}$

$8 x^{3}$

$16 x^{2}$

$19 x^{2}$

$2 x$

Step 1.13

Multiply the new quotient term by the divisor.

$3 x^{3}$

$8 x^{2}$

$19 x$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

$8 x^{3}$

$3 x^{2}$

$8 x^{3}$

$16 x^{2}$

$19 x^{2}$

$2 x$

$19 x^{2}$

$38 x$

Step 1.14

The expression needs to be subtracted from the dividend, so change all the signs in $19 x^{2} + 38 x$

$3 x^{3}$

$8 x^{2}$

$19 x$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

$8 x^{3}$

$3 x^{2}$

$8 x^{3}$

$16 x^{2}$

$19 x^{2}$

$2 x$

$19 x^{2}$

$38 x$

Step 1.15

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

$3 x^{3}$

$8 x^{2}$

$19 x$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

$8 x^{3}$

$3 x^{2}$

$8 x^{3}$

$16 x^{2}$

$19 x^{2}$

$2 x$

$19 x^{2}$

$38 x$

$40 x$

Step 1.16

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

$3 x^{3}$

$8 x^{2}$

$19 x$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

$8 x^{3}$

$3 x^{2}$

$8 x^{3}$

$16 x^{2}$

$19 x^{2}$

$2 x$

$19 x^{2}$

$38 x$

$40 x$

$3$

Step 1.17

Divide the highest order term in the dividend $- 40 x$ by the highest order term in divisor $x$ .

$3 x^{3}$

$8 x^{2}$

$19 x$

$40$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

$8 x^{3}$

$3 x^{2}$

$8 x^{3}$

$16 x^{2}$

$19 x^{2}$

$2 x$

$19 x^{2}$

$38 x$

$40 x$

$3$

Step 1.18

Multiply the new quotient term by the divisor.

$3 x^{3}$

$8 x^{2}$

$19 x$

$40$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

$8 x^{3}$

$3 x^{2}$

$8 x^{3}$

$16 x^{2}$

$19 x^{2}$

$2 x$

$19 x^{2}$

$38 x$

$40 x$

$3$

$40 x$

$80$

Step 1.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 40 x - 80$

$3 x^{3}$

$8 x^{2}$

$19 x$

$40$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

$8 x^{3}$

$3 x^{2}$

$8 x^{3}$

$16 x^{2}$

$19 x^{2}$

$2 x$

$19 x^{2}$

$38 x$

$40 x$

$3$

$40 x$

$80$

Step 1.20

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

$3 x^{3}$

$8 x^{2}$

$19 x$

$40$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

$3 x^{2}$

$2 x$

$3$

$3 x^{4}$

$6 x^{3}$

$8 x^{3}$

$3 x^{2}$

$8 x^{3}$

$16 x^{2}$

$19 x^{2}$

$2 x$

$19 x^{2}$

$38 x$

$40 x$

$3$

$40 x$

$80$

$83$

Step 1.21

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

$3 x^{3} - 8 x^{2} + 19 x - 40 + \frac{83}{x + 2}$

Step 2

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

$83$