Find the Remainder (15x^4+22x^3-17x^2+4x+12)/(x+2)

$\frac{15 x^{4} + 22 x^{3} - 17 x^{2} + 4 x + 12}{x + 2}$

Step 1

To calculate the remainder, first divide the polynomials.

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

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

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

$15 x^{3}$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

Multiply the new quotient term by the divisor.

$15 x^{3}$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $15 x^{4} + 30 x^{3}$

$15 x^{3}$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

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

$15 x^{3}$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

$8 x^{3}$

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

$15 x^{3}$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

$8 x^{3}$

$17 x^{2}$

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

$15 x^{3}$

$8 x^{2}$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

$8 x^{3}$

$17 x^{2}$

Multiply the new quotient term by the divisor.

$15 x^{3}$

$8 x^{2}$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

$8 x^{3}$

$17 x^{2}$

$8 x^{3}$

$16 x^{2}$

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

$15 x^{3}$

$8 x^{2}$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

$8 x^{3}$

$17 x^{2}$

$8 x^{3}$

$16 x^{2}$

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

$15 x^{3}$

$8 x^{2}$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

$8 x^{3}$

$17 x^{2}$

$8 x^{3}$

$16 x^{2}$

$x^{2}$

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

$15 x^{3}$

$8 x^{2}$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

$8 x^{3}$

$17 x^{2}$

$8 x^{3}$

$16 x^{2}$

$x^{2}$

$4 x$

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

$15 x^{3}$

$8 x^{2}$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

$8 x^{3}$

$17 x^{2}$

$8 x^{3}$

$16 x^{2}$

$x^{2}$

$4 x$

Multiply the new quotient term by the divisor.

$15 x^{3}$

$8 x^{2}$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

$8 x^{3}$

$17 x^{2}$

$8 x^{3}$

$16 x^{2}$

$x^{2}$

$4 x$

$x^{2}$

$2 x$

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

$15 x^{3}$

$8 x^{2}$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

$8 x^{3}$

$17 x^{2}$

$8 x^{3}$

$16 x^{2}$

$x^{2}$

$4 x$

$x^{2}$

$2 x$

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

$15 x^{3}$

$8 x^{2}$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

$8 x^{3}$

$17 x^{2}$

$8 x^{3}$

$16 x^{2}$

$x^{2}$

$4 x$

$x^{2}$

$2 x$

$6 x$

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

$15 x^{3}$

$8 x^{2}$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

$8 x^{3}$

$17 x^{2}$

$8 x^{3}$

$16 x^{2}$

$x^{2}$

$4 x$

$x^{2}$

$2 x$

$6 x$

$12$

Divide the highest order term in the dividend $6 x$ by the highest order term in divisor $x$ .

$15 x^{3}$

$8 x^{2}$

$x$

$6$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

$8 x^{3}$

$17 x^{2}$

$8 x^{3}$

$16 x^{2}$

$x^{2}$

$4 x$

$x^{2}$

$2 x$

$6 x$

$12$

Multiply the new quotient term by the divisor.

$15 x^{3}$

$8 x^{2}$

$x$

$6$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

$8 x^{3}$

$17 x^{2}$

$8 x^{3}$

$16 x^{2}$

$x^{2}$

$4 x$

$x^{2}$

$2 x$

$6 x$

$12$

$6 x$

$12$

The expression needs to be subtracted from the dividend, so change all the signs in $6 x + 12$

$15 x^{3}$

$8 x^{2}$

$x$

$6$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

$8 x^{3}$

$17 x^{2}$

$8 x^{3}$

$16 x^{2}$

$x^{2}$

$4 x$

$x^{2}$

$2 x$

$6 x$

$12$

$6 x$

$12$

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

$15 x^{3}$

$8 x^{2}$

$x$

$6$

$x$

$2$

$15 x^{4}$

$22 x^{3}$

$17 x^{2}$

$4 x$

$12$

$15 x^{4}$

$30 x^{3}$

$8 x^{3}$

$17 x^{2}$

$8 x^{3}$

$16 x^{2}$

$x^{2}$

$4 x$

$x^{2}$

$2 x$

$6 x$

$12$

$6 x$

$12$

$0$

Since the remander is $0$ , the final answer is the quotient.

$15 x^{3} - 8 x^{2} - x + 6$

Step 2

Since the final term in the resulting expression is not a fraction, the remainder is $0$ .

$0$

Find the Remainder (15x^4+22x^3-17x^2+4x+12)/(x+2)

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