Find the Remainder (3x^4+2x^3+4x)÷(x+5)

$(3 x^{4} + 2 x^{3} + 4 x) \div (x + 5)$

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 $0$ .

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

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

$3 x^{3}$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

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

$3 x^{3}$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

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

$3 x^{3}$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

$13 x^{3}$

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

$3 x^{3}$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

$13 x^{3}$

$0 x^{2}$

Divide the highest order term in the dividend $- 13 x^{3}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$13 x^{2}$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

$13 x^{3}$

$0 x^{2}$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$13 x^{2}$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

$13 x^{3}$

$0 x^{2}$

$13 x^{3}$

$65 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $- 13 x^{3} - 65 x^{2}$

$3 x^{3}$

$13 x^{2}$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

$13 x^{3}$

$0 x^{2}$

$13 x^{3}$

$65 x^{2}$

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

$3 x^{3}$

$13 x^{2}$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

$13 x^{3}$

$0 x^{2}$

$13 x^{3}$

$65 x^{2}$

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

$3 x^{3}$

$13 x^{2}$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

$13 x^{3}$

$0 x^{2}$

$13 x^{3}$

$65 x^{2}$

$4 x$

Divide the highest order term in the dividend $65 x^{2}$ by the highest order term in divisor $x$ .

$3 x^{3}$

$13 x^{2}$

$65 x$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

$13 x^{3}$

$0 x^{2}$

$13 x^{3}$

$65 x^{2}$

$4 x$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$13 x^{2}$

$65 x$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

$13 x^{3}$

$0 x^{2}$

$13 x^{3}$

$65 x^{2}$

$4 x$

$65 x^{2}$

$325 x$

The expression needs to be subtracted from the dividend, so change all the signs in $65 x^{2} + 325 x$

$3 x^{3}$

$13 x^{2}$

$65 x$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

$13 x^{3}$

$0 x^{2}$

$13 x^{3}$

$65 x^{2}$

$4 x$

$65 x^{2}$

$325 x$

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

$3 x^{3}$

$13 x^{2}$

$65 x$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

$13 x^{3}$

$0 x^{2}$

$13 x^{3}$

$65 x^{2}$

$4 x$

$65 x^{2}$

$325 x$

$321 x$

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

$3 x^{3}$

$13 x^{2}$

$65 x$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

$13 x^{3}$

$0 x^{2}$

$13 x^{3}$

$65 x^{2}$

$4 x$

$65 x^{2}$

$325 x$

$321 x$

$0$

Divide the highest order term in the dividend $- 321 x$ by the highest order term in divisor $x$ .

$3 x^{3}$

$13 x^{2}$

$65 x$

$321$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

$13 x^{3}$

$0 x^{2}$

$13 x^{3}$

$65 x^{2}$

$4 x$

$65 x^{2}$

$325 x$

$321 x$

$0$

Multiply the new quotient term by the divisor.

$3 x^{3}$

$13 x^{2}$

$65 x$

$321$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

$13 x^{3}$

$0 x^{2}$

$13 x^{3}$

$65 x^{2}$

$4 x$

$65 x^{2}$

$325 x$

$321 x$

$0$

$321 x$

$1605$

The expression needs to be subtracted from the dividend, so change all the signs in $- 321 x - 1605$

$3 x^{3}$

$13 x^{2}$

$65 x$

$321$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

$13 x^{3}$

$0 x^{2}$

$13 x^{3}$

$65 x^{2}$

$4 x$

$65 x^{2}$

$325 x$

$321 x$

$0$

$321 x$

$1605$

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

$3 x^{3}$

$13 x^{2}$

$65 x$

$321$

$x$

$5$

$3 x^{4}$

$2 x^{3}$

$0 x^{2}$

$4 x$

$0$

$3 x^{4}$

$15 x^{3}$

$13 x^{3}$

$0 x^{2}$

$13 x^{3}$

$65 x^{2}$

$4 x$

$65 x^{2}$

$325 x$

$321 x$

$0$

$321 x$

$1605$

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

$3 x^{3} - 13 x^{2} + 65 x - 321 + \frac{1605}{x + 5}$

Step 2

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

$1605$

Find the Remainder (3x^4+2x^3+4x)÷(x+5)

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