Algebra Examples

$\frac{- 8 x^{5} + 10 x^{3} - 100}{x + 2}$

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 $0$ .

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

Divide the highest order term in the dividend $- 8 x^{5}$ by the highest order term in divisor $x$ .

$8 x^{4}$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

Multiply the new quotient term by the divisor.

$8 x^{4}$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 8 x^{5} - 16 x^{4}$

$8 x^{4}$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

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

$8 x^{4}$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

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

$8 x^{4}$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

Divide the highest order term in the dividend $16 x^{4}$ by the highest order term in divisor $x$ .

$8 x^{4}$

$16 x^{3}$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

Multiply the new quotient term by the divisor.

$8 x^{4}$

$16 x^{3}$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $16 x^{4} + 32 x^{3}$

$8 x^{4}$

$16 x^{3}$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

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

$8 x^{4}$

$16 x^{3}$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

$22 x^{3}$

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

$8 x^{4}$

$16 x^{3}$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

$22 x^{3}$

$0 x^{2}$

Divide the highest order term in the dividend $- 22 x^{3}$ by the highest order term in divisor $x$ .

$8 x^{4}$

$16 x^{3}$

$22 x^{2}$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

$22 x^{3}$

$0 x^{2}$

Multiply the new quotient term by the divisor.

$8 x^{4}$

$16 x^{3}$

$22 x^{2}$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$44 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 22 x^{3} - 44 x^{2}$

$8 x^{4}$

$16 x^{3}$

$22 x^{2}$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$44 x^{2}$

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

$8 x^{4}$

$16 x^{3}$

$22 x^{2}$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$44 x^{2}$

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

$8 x^{4}$

$16 x^{3}$

$22 x^{2}$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$44 x^{2}$

$0 x$

Divide the highest order term in the dividend $44 x^{2}$ by the highest order term in divisor $x$ .

$8 x^{4}$

$16 x^{3}$

$22 x^{2}$

$44 x$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$44 x^{2}$

$0 x$

Multiply the new quotient term by the divisor.

$8 x^{4}$

$16 x^{3}$

$22 x^{2}$

$44 x$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$44 x^{2}$

$0 x$

$44 x^{2}$

$88 x$

The expression needs to be subtracted from the dividend, so change all the signs in $44 x^{2} + 88 x$

$8 x^{4}$

$16 x^{3}$

$22 x^{2}$

$44 x$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$44 x^{2}$

$0 x$

$44 x^{2}$

$88 x$

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

$8 x^{4}$

$16 x^{3}$

$22 x^{2}$

$44 x$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$44 x^{2}$

$0 x$

$44 x^{2}$

$88 x$

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

$8 x^{4}$

$16 x^{3}$

$22 x^{2}$

$44 x$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$44 x^{2}$

$0 x$

$44 x^{2}$

$88 x$

$100$

Divide the highest order term in the dividend $- 88 x$ by the highest order term in divisor $x$ .

$8 x^{4}$

$16 x^{3}$

$22 x^{2}$

$44 x$

$88$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$44 x^{2}$

$0 x$

$44 x^{2}$

$88 x$

$100$

Multiply the new quotient term by the divisor.

$8 x^{4}$

$16 x^{3}$

$22 x^{2}$

$44 x$

$88$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$44 x^{2}$

$0 x$

$44 x^{2}$

$88 x$

$100$

$88 x$

$176$

The expression needs to be subtracted from the dividend, so change all the signs in $- 88 x - 176$

$8 x^{4}$

$16 x^{3}$

$22 x^{2}$

$44 x$

$88$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$44 x^{2}$

$0 x$

$44 x^{2}$

$88 x$

$100$

$88 x$

$176$

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

$8 x^{4}$

$16 x^{3}$

$22 x^{2}$

$44 x$

$88$

$x$

$2$

$8 x^{5}$

$0 x^{4}$

$10 x^{3}$

$0 x^{2}$

$0 x$

$100$

$8 x^{5}$

$16 x^{4}$

$10 x^{3}$

$16 x^{4}$

$32 x^{3}$

$22 x^{3}$

$0 x^{2}$

$22 x^{3}$

$44 x^{2}$

$0 x$

$44 x^{2}$

$88 x$

$100$

$88 x$

$176$

$76$

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

$- 8 x^{4} + 16 x^{3} - 22 x^{2} + 44 x - 88 + \frac{76}{x + 2}$

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

$76$