Find the Remainder (3x^4+2x^3-x^2+2x-24)/(x+2)

$\frac{(3 x^{4} + 2 x^{3} - x^{2} + 2 x - 24)}{(x + 2)}$

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 $0$ .

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

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

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

$3 x^{4}$

$6 x^{3}$

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

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

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

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

–

$4 x^{3}$

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

$3 x^{3}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

–

$4 x^{3}$

–

$x^{2}$

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

$3 x^{3}$

–

$4 x^{2}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

–

$4 x^{3}$

–

$x^{2}$

Multiply the new quotient term by the divisor.

$3 x^{3}$

–

$4 x^{2}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

–

$4 x^{3}$

–

$x^{2}$

–

$4 x^{3}$

–

$8 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{3} - 8 x^{2}$

$3 x^{3}$

–

$4 x^{2}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

–

$4 x^{3}$

–

$x^{2}$

$4 x^{3}$

$8 x^{2}$

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

$3 x^{3}$

–

$4 x^{2}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

–

$4 x^{3}$

–

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

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

$3 x^{3}$

–

$4 x^{2}$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

–

$4 x^{3}$

–

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

Divide the highest order term in the dividend $7 x^{2}$ by the highest order term in divisor $x$ .

$3 x^{3}$

–

$4 x^{2}$

$7 x$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

–

$4 x^{3}$

–

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

Multiply the new quotient term by the divisor.

$3 x^{3}$

–

$4 x^{2}$

$7 x$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

–

$4 x^{3}$

–

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

$7 x^{2}$

$14 x$

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{2} + 14 x$

$3 x^{3}$

–

$4 x^{2}$

$7 x$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

–

$4 x^{3}$

–

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

–

$7 x^{2}$

–

$14 x$

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

$3 x^{3}$

–

$4 x^{2}$

$7 x$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

–

$4 x^{3}$

–

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

–

$7 x^{2}$

–

$14 x$

–

$12 x$

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

$3 x^{3}$

–

$4 x^{2}$

$7 x$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

–

$4 x^{3}$

–

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

–

$7 x^{2}$

–

$14 x$

–

$12 x$

–

$24$

Divide the highest order term in the dividend $- 12 x$ by the highest order term in divisor $x$ .

$3 x^{3}$

–

$4 x^{2}$

$7 x$

–

$12$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

–

$4 x^{3}$

–

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

–

$7 x^{2}$

–

$14 x$

–

$12 x$

–

$24$

Multiply the new quotient term by the divisor.

$3 x^{3}$

–

$4 x^{2}$

$7 x$

–

$12$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

–

$4 x^{3}$

–

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

–

$7 x^{2}$

–

$14 x$

–

$12 x$

–

$24$

–

$12 x$

–

$24$

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x - 24$

$3 x^{3}$

–

$4 x^{2}$

$7 x$

–

$12$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

–

$4 x^{3}$

–

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

–

$7 x^{2}$

–

$14 x$

–

$12 x$

–

$24$

$12 x$

$24$

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

$3 x^{3}$

–

$4 x^{2}$

$7 x$

–

$12$

$x$

$2$

$3 x^{4}$

$2 x^{3}$

–

$x^{2}$

$2 x$

–

$24$

–

$3 x^{4}$

–

$6 x^{3}$

–

$4 x^{3}$

–

$x^{2}$

$4 x^{3}$

$8 x^{2}$

$7 x^{2}$

$2 x$

–

$7 x^{2}$

–

$14 x$

–

$12 x$

–

$24$

$12 x$

$24$

$0$

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

$3 x^{3} - 4 x^{2} + 7 x - 12$

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

$0$

Find the Remainder (3x^4+2x^3-x^2+2x-24)/(x+2)

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