Algebra Examples

$(- 2 x^{4} + 6 x^{2} - 15 + 3 x^{3}) \div (- x^{2} + x + 2)$

Step 1

To calculate the remainder, first divide the polynomials.

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Step 1.1

Expand $- 2 x^{4} + 6 x^{2} - 15 + 3 x^{3}$ .

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Step 1.1.1

Move $- 15$ .

$\frac{- 2 x^{4} + 6 x^{2} + 3 x^{3} - 15}{- x^{2} + x + 2}$

Step 1.1.2

Move $6 x^{2}$ .

$\frac{- 2 x^{4} + 3 x^{3} + 6 x^{2} - 15}{- x^{2} + x + 2}$

Step 1.2

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$x^{2}$

$x$

$2$

$2 x^{4}$

$3 x^{3}$

$6 x^{2}$

$0 x$

$15$

Step 1.3

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

$2 x^{2}$

$x^{2}$

$x$

$2$

$2 x^{4}$

$3 x^{3}$

$6 x^{2}$

$0 x$

$15$

Step 1.4

Multiply the new quotient term by the divisor.

$2 x^{2}$

$x^{2}$

$x$

$2$

$2 x^{4}$

$3 x^{3}$

$6 x^{2}$

$0 x$

$15$

$2 x^{4}$

$2 x^{3}$

$4 x^{2}$

Step 1.5

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

$2 x^{2}$

$x^{2}$

$x$

$2$

$2 x^{4}$

$3 x^{3}$

$6 x^{2}$

$0 x$

$15$

$2 x^{4}$

$2 x^{3}$

$4 x^{2}$

Step 1.6

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

$2 x^{2}$

$x^{2}$

$x$

$2$

$2 x^{4}$

$3 x^{3}$

$6 x^{2}$

$0 x$

$15$

$2 x^{4}$

$2 x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

Step 1.7

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

$2 x^{2}$

$x^{2}$

$x$

$2$

$2 x^{4}$

$3 x^{3}$

$6 x^{2}$

$0 x$

$15$

$2 x^{4}$

$2 x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

Step 1.8

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

$2 x^{2}$

$x$

$x^{2}$

$x$

$2$

$2 x^{4}$

$3 x^{3}$

$6 x^{2}$

$0 x$

$15$

$2 x^{4}$

$2 x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

Step 1.9

Multiply the new quotient term by the divisor.

$2 x^{2}$

$x$

$x^{2}$

$x$

$2$

$2 x^{4}$

$3 x^{3}$

$6 x^{2}$

$0 x$

$15$

$2 x^{4}$

$2 x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

$x^{3}$

$x^{2}$

$2 x$

Step 1.10

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

$2 x^{2}$

$x$

$x^{2}$

$x$

$2$

$2 x^{4}$

$3 x^{3}$

$6 x^{2}$

$0 x$

$15$

$2 x^{4}$

$2 x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

$x^{3}$

$x^{2}$

$2 x$

Step 1.11

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

$2 x^{2}$

$x$

$x^{2}$

$x$

$2$

$2 x^{4}$

$3 x^{3}$

$6 x^{2}$

$0 x$

$15$

$2 x^{4}$

$2 x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

$x^{3}$

$x^{2}$

$2 x$

$3 x^{2}$

$2 x$

Step 1.12

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

$2 x^{2}$

$x$

$x^{2}$

$x$

$2$

$2 x^{4}$

$3 x^{3}$

$6 x^{2}$

$0 x$

$15$

$2 x^{4}$

$2 x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

$x^{3}$

$x^{2}$

$2 x$

$3 x^{2}$

$2 x$

$15$

Step 1.13

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

$2 x^{2}$

$x$

$3$

$x^{2}$

$x$

$2$

$2 x^{4}$

$3 x^{3}$

$6 x^{2}$

$0 x$

$15$

$2 x^{4}$

$2 x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

$x^{3}$

$x^{2}$

$2 x$

$3 x^{2}$

$2 x$

$15$

Step 1.14

Multiply the new quotient term by the divisor.

$2 x^{2}$

$x$

$3$

$x^{2}$

$x$

$2$

$2 x^{4}$

$3 x^{3}$

$6 x^{2}$

$0 x$

$15$

$2 x^{4}$

$2 x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

$x^{3}$

$x^{2}$

$2 x$

$3 x^{2}$

$2 x$

$15$

$3 x^{2}$

$3 x$

$6$

Step 1.15

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

$2 x^{2}$

$x$

$3$

$x^{2}$

$x$

$2$

$2 x^{4}$

$3 x^{3}$

$6 x^{2}$

$0 x$

$15$

$2 x^{4}$

$2 x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

$x^{3}$

$x^{2}$

$2 x$

$3 x^{2}$

$2 x$

$15$

$3 x^{2}$

$3 x$

$6$

Step 1.16

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

$2 x^{2}$

$x$

$3$

$x^{2}$

$x$

$2$

$2 x^{4}$

$3 x^{3}$

$6 x^{2}$

$0 x$

$15$

$2 x^{4}$

$2 x^{3}$

$4 x^{2}$

$x^{3}$

$2 x^{2}$

$0 x$

$x^{3}$

$x^{2}$

$2 x$

$3 x^{2}$

$2 x$

$15$

$3 x^{2}$

$3 x$

$6$

$5 x$

$9$

Step 1.17

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

$2 x^{2} - x - 3 + \frac{5 x - 9}{- x^{2} + x + 2}$

Step 2

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

$5 x - 9$