Divide Using Long Polynomial Division (3x^4+7x^3+2x^2+13x+5)÷(x^2+3x+1)

$(3 x^{4} + 7 x^{3} + 2 x^{2} + 13 x + 5) \div (x^{2} + 3 x + 1)$

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

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$13 x$

$5$

Divide the highest order term in the dividend $3 x^{4}$ by the highest order term in divisor $x^{2}$ .

$3 x^{2}$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$13 x$

$5$

Multiply the new quotient term by the divisor.

$3 x^{2}$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$13 x$

$5$

$3 x^{4}$

$9 x^{3}$

$3 x^{2}$

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{4} + 9 x^{3} + 3 x^{2}$

$3 x^{2}$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$13 x$

$5$

–

$3 x^{4}$

–

$9 x^{3}$

–

$3 x^{2}$

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

$3 x^{2}$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$13 x$

$5$

–

$3 x^{4}$

–

$9 x^{3}$

–

$3 x^{2}$

–

$2 x^{3}$

–

$x^{2}$

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

$3 x^{2}$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$13 x$

$5$

–

$3 x^{4}$

–

$9 x^{3}$

–

$3 x^{2}$

–

$2 x^{3}$

–

$x^{2}$

$13 x$

Divide the highest order term in the dividend $- 2 x^{3}$ by the highest order term in divisor $x^{2}$ .

$3 x^{2}$

–

$2 x$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$13 x$

$5$

–

$3 x^{4}$

–

$9 x^{3}$

–

$3 x^{2}$

–

$2 x^{3}$

–

$x^{2}$

$13 x$

Multiply the new quotient term by the divisor.

$3 x^{2}$

–

$2 x$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$13 x$

$5$

–

$3 x^{4}$

–

$9 x^{3}$

–

$3 x^{2}$

–

$2 x^{3}$

–

$x^{2}$

$13 x$

–

$2 x^{3}$

–

$6 x^{2}$

–

$2 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{3} - 6 x^{2} - 2 x$

$3 x^{2}$

–

$2 x$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$13 x$

$5$

–

$3 x^{4}$

–

$9 x^{3}$

–

$3 x^{2}$

–

$2 x^{3}$

–

$x^{2}$

$13 x$

$2 x^{3}$

$6 x^{2}$

$2 x$

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

$3 x^{2}$

–

$2 x$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$13 x$

$5$

–

$3 x^{4}$

–

$9 x^{3}$

–

$3 x^{2}$

–

$2 x^{3}$

–

$x^{2}$

$13 x$

$2 x^{3}$

$6 x^{2}$

$2 x$

$5 x^{2}$

$15 x$

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

$3 x^{2}$

–

$2 x$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$13 x$

$5$

–

$3 x^{4}$

–

$9 x^{3}$

–

$3 x^{2}$

–

$2 x^{3}$

–

$x^{2}$

$13 x$

$2 x^{3}$

$6 x^{2}$

$2 x$

$5 x^{2}$

$15 x$

$5$

Divide the highest order term in the dividend $5 x^{2}$ by the highest order term in divisor $x^{2}$ .

$3 x^{2}$

–

$2 x$

$5$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$13 x$

$5$

–

$3 x^{4}$

–

$9 x^{3}$

–

$3 x^{2}$

–

$2 x^{3}$

–

$x^{2}$

$13 x$

$2 x^{3}$

$6 x^{2}$

$2 x$

$5 x^{2}$

$15 x$

$5$

Multiply the new quotient term by the divisor.

$3 x^{2}$

–

$2 x$

$5$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$13 x$

$5$

–

$3 x^{4}$

–

$9 x^{3}$

–

$3 x^{2}$

–

$2 x^{3}$

–

$x^{2}$

$13 x$

$2 x^{3}$

$6 x^{2}$

$2 x$

$5 x^{2}$

$15 x$

$5$

$5 x^{2}$

$15 x$

$5$

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{2} + 15 x + 5$

$3 x^{2}$

–

$2 x$

$5$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$13 x$

$5$

–

$3 x^{4}$

–

$9 x^{3}$

–

$3 x^{2}$

–

$2 x^{3}$

–

$x^{2}$

$13 x$

$2 x^{3}$

$6 x^{2}$

$2 x$

$5 x^{2}$

$15 x$

$5$

–

$5 x^{2}$

–

$15 x$

–

$5$

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

$3 x^{2}$

–

$2 x$

$5$

$x^{2}$

$3 x$

$1$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$13 x$

$5$

–

$3 x^{4}$

–

$9 x^{3}$

–

$3 x^{2}$

–

$2 x^{3}$

–

$x^{2}$

$13 x$

$2 x^{3}$

$6 x^{2}$

$2 x$

$5 x^{2}$

$15 x$

$5$

–

$5 x^{2}$

–

$15 x$

–

$5$

$0$

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

$3 x^{2} - 2 x + 5$

Divide Using Long Polynomial Division (3x^4+7x^3+2x^2+13x+5)÷(x^2+3x+1)

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