Find the Remainder (7x^4+x+14)/(x+2)

$\frac{7 x^{4} + x + 14}{x + 2}$

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 $0$ .

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

Divide the highest order term in the dividend $7 x^{4}$ by the highest order term in divisor $x$ .

$7 x^{3}$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

Multiply the new quotient term by the divisor.

$7 x^{3}$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{4} + 14 x^{3}$

$7 x^{3}$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

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

$7 x^{3}$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

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

$7 x^{3}$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

$0 x^{2}$

Divide the highest order term in the dividend $- 14 x^{3}$ by the highest order term in divisor $x$ .

$7 x^{3}$

$14 x^{2}$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

$0 x^{2}$

Multiply the new quotient term by the divisor.

$7 x^{3}$

$14 x^{2}$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

$0 x^{2}$

$14 x^{3}$

$28 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 14 x^{3} - 28 x^{2}$

$7 x^{3}$

$14 x^{2}$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

$0 x^{2}$

$14 x^{3}$

$28 x^{2}$

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

$7 x^{3}$

$14 x^{2}$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

$0 x^{2}$

$14 x^{3}$

$28 x^{2}$

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

$7 x^{3}$

$14 x^{2}$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

$0 x^{2}$

$14 x^{3}$

$28 x^{2}$

$x$

Divide the highest order term in the dividend $28 x^{2}$ by the highest order term in divisor $x$ .

$7 x^{3}$

$14 x^{2}$

$28 x$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

$0 x^{2}$

$14 x^{3}$

$28 x^{2}$

$x$

Multiply the new quotient term by the divisor.

$7 x^{3}$

$14 x^{2}$

$28 x$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

$0 x^{2}$

$14 x^{3}$

$28 x^{2}$

$x$

$28 x^{2}$

$56 x$

The expression needs to be subtracted from the dividend, so change all the signs in $28 x^{2} + 56 x$

$7 x^{3}$

$14 x^{2}$

$28 x$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

$0 x^{2}$

$14 x^{3}$

$28 x^{2}$

$x$

$28 x^{2}$

$56 x$

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

$7 x^{3}$

$14 x^{2}$

$28 x$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

$0 x^{2}$

$14 x^{3}$

$28 x^{2}$

$x$

$28 x^{2}$

$56 x$

$55 x$

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

$7 x^{3}$

$14 x^{2}$

$28 x$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

$0 x^{2}$

$14 x^{3}$

$28 x^{2}$

$x$

$28 x^{2}$

$56 x$

$55 x$

$14$

Divide the highest order term in the dividend $- 55 x$ by the highest order term in divisor $x$ .

$7 x^{3}$

$14 x^{2}$

$28 x$

$55$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

$0 x^{2}$

$14 x^{3}$

$28 x^{2}$

$x$

$28 x^{2}$

$56 x$

$55 x$

$14$

Multiply the new quotient term by the divisor.

$7 x^{3}$

$14 x^{2}$

$28 x$

$55$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

$0 x^{2}$

$14 x^{3}$

$28 x^{2}$

$x$

$28 x^{2}$

$56 x$

$55 x$

$14$

$55 x$

$110$

The expression needs to be subtracted from the dividend, so change all the signs in $- 55 x - 110$

$7 x^{3}$

$14 x^{2}$

$28 x$

$55$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

$0 x^{2}$

$14 x^{3}$

$28 x^{2}$

$x$

$28 x^{2}$

$56 x$

$55 x$

$14$

$55 x$

$110$

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

$7 x^{3}$

$14 x^{2}$

$28 x$

$55$

$x$

$2$

$7 x^{4}$

$0 x^{3}$

$0 x^{2}$

$x$

$14$

$7 x^{4}$

$14 x^{3}$

$0 x^{2}$

$14 x^{3}$

$28 x^{2}$

$x$

$28 x^{2}$

$56 x$

$55 x$

$14$

$55 x$

$110$

$124$

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

$7 x^{3} - 14 x^{2} + 28 x - 55 + \frac{124}{x + 2}$

Step 2

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

$124$

Find the Remainder (7x^4+x+14)/(x+2)

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