Find the Remainder (2x^6+x^2+2)/(x+2)

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Algebra

$\frac{2 x^{6} + x^{2} + 2}{x + 2}$

To calculate the remainder, first divide the polynomials.

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

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

Divide the highest order term in the dividend $2 x^{6}$ by the highest order term in divisor $x$ .

$2 x^{5}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

Multiply the new quotient term by the divisor.

$2 x^{5}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{6} + 4 x^{5}$

$2 x^{5}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

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

$2 x^{5}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

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

$2 x^{5}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

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

$2 x^{5}$

$4 x^{4}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

Multiply the new quotient term by the divisor.

$2 x^{5}$

$4 x^{4}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

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

$2 x^{5}$

$4 x^{4}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

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

$2 x^{5}$

$4 x^{4}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

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

$2 x^{5}$

$4 x^{4}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

Divide the highest order term in the dividend $8 x^{4}$ by the highest order term in divisor $x$ .

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

Multiply the new quotient term by the divisor.

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{4} + 16 x^{3}$

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

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

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

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

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

$x^{2}$

Divide the highest order term in the dividend $- 16 x^{3}$ by the highest order term in divisor $x$ .

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$16 x^{2}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

$x^{2}$

Multiply the new quotient term by the divisor.

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$16 x^{2}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

$x^{2}$

$16 x^{3}$

$32 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 16 x^{3} - 32 x^{2}$

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$16 x^{2}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

$x^{2}$

$16 x^{3}$

$32 x^{2}$

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

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$16 x^{2}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

$x^{2}$

$16 x^{3}$

$32 x^{2}$

$33 x^{2}$

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

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$16 x^{2}$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

$x^{2}$

$16 x^{3}$

$32 x^{2}$

$33 x^{2}$

$0 x$

Divide the highest order term in the dividend $33 x^{2}$ by the highest order term in divisor $x$ .

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$16 x^{2}$

$33 x$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

$x^{2}$

$16 x^{3}$

$32 x^{2}$

$33 x^{2}$

$0 x$

Multiply the new quotient term by the divisor.

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$16 x^{2}$

$33 x$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

$x^{2}$

$16 x^{3}$

$32 x^{2}$

$33 x^{2}$

$0 x$

$33 x^{2}$

$66 x$

The expression needs to be subtracted from the dividend, so change all the signs in $33 x^{2} + 66 x$

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$16 x^{2}$

$33 x$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

$x^{2}$

$16 x^{3}$

$32 x^{2}$

$33 x^{2}$

$0 x$

$33 x^{2}$

$66 x$

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

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$16 x^{2}$

$33 x$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

$x^{2}$

$16 x^{3}$

$32 x^{2}$

$33 x^{2}$

$0 x$

$33 x^{2}$

$66 x$

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

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$16 x^{2}$

$33 x$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

$x^{2}$

$16 x^{3}$

$32 x^{2}$

$33 x^{2}$

$0 x$

$33 x^{2}$

$66 x$

$2$

Divide the highest order term in the dividend $- 66 x$ by the highest order term in divisor $x$ .

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$16 x^{2}$

$33 x$

$66$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

$x^{2}$

$16 x^{3}$

$32 x^{2}$

$33 x^{2}$

$0 x$

$33 x^{2}$

$66 x$

$2$

Multiply the new quotient term by the divisor.

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$16 x^{2}$

$33 x$

$66$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

$x^{2}$

$16 x^{3}$

$32 x^{2}$

$33 x^{2}$

$0 x$

$33 x^{2}$

$66 x$

$2$

$66 x$

$132$

The expression needs to be subtracted from the dividend, so change all the signs in $- 66 x - 132$

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$16 x^{2}$

$33 x$

$66$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

$x^{2}$

$16 x^{3}$

$32 x^{2}$

$33 x^{2}$

$0 x$

$33 x^{2}$

$66 x$

$2$

$66 x$

$132$

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

$2 x^{5}$

$4 x^{4}$

$8 x^{3}$

$16 x^{2}$

$33 x$

$66$

$x$

$2$

$2 x^{6}$

$0 x^{5}$

$0 x^{4}$

$0 x^{3}$

$x^{2}$

$0 x$

$2$

$2 x^{6}$

$4 x^{5}$

$0 x^{4}$

$4 x^{5}$

$8 x^{4}$

$0 x^{3}$

$8 x^{4}$

$16 x^{3}$

$x^{2}$

$16 x^{3}$

$32 x^{2}$

$33 x^{2}$

$0 x$

$33 x^{2}$

$66 x$

$2$

$66 x$

$132$

$134$

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

$2 x^{5} - 4 x^{4} + 8 x^{3} - 16 x^{2} + 33 x - 66 + \frac{134}{x + 2}$

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

$134$

Do you know how to Find the Remainder (2x^6+x^2+2)/(x+2)? If not, you can write to our math experts in our application. The best solution for your task you can find above on this page.

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