Divide Using Long Polynomial Division (x^4+x^3-5x^2+13x-6)/(x^2+3x-2)

$\frac{x^{4} + x^{3} - 5 x^{2} + 13 x - 6}{x^{2} + 3 x - 2}$

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

$x^{2}$

$3 x$

$2$

$x^{4}$

$x^{3}$

$5 x^{2}$

$13 x$

$6$

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

$x^{2}$

$3 x$

$2$

$x^{4}$

$x^{3}$

$5 x^{2}$

$13 x$

$6$

Multiply the new quotient term by the divisor.

$x^{2}$

$3 x$

$2$

$x^{4}$

$x^{3}$

$5 x^{2}$

$13 x$

$6$

$x^{4}$

$3 x^{3}$

$2 x^{2}$

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

$x^{2}$

$3 x$

$2$

$x^{4}$

$x^{3}$

$5 x^{2}$

$13 x$

$6$

$x^{4}$

$3 x^{3}$

$2 x^{2}$

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

$x^{2}$

$3 x$

$2$

$x^{4}$

$x^{3}$

$5 x^{2}$

$13 x$

$6$

$x^{4}$

$3 x^{3}$

$2 x^{2}$

$2 x^{3}$

$3 x^{2}$

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

$x^{2}$

$3 x$

$2$

$x^{4}$

$x^{3}$

$5 x^{2}$

$13 x$

$6$

$x^{4}$

$3 x^{3}$

$2 x^{2}$

$2 x^{3}$

$3 x^{2}$

$13 x$

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

$x^{2}$

$2 x$

$x^{2}$

$3 x$

$2$

$x^{4}$

$x^{3}$

$5 x^{2}$

$13 x$

$6$

$x^{4}$

$3 x^{3}$

$2 x^{2}$

$2 x^{3}$

$3 x^{2}$

$13 x$

Multiply the new quotient term by the divisor.

$x^{2}$

$2 x$

$x^{2}$

$3 x$

$2$

$x^{4}$

$x^{3}$

$5 x^{2}$

$13 x$

$6$

$x^{4}$

$3 x^{3}$

$2 x^{2}$

$2 x^{3}$

$3 x^{2}$

$13 x$

$2 x^{3}$

$6 x^{2}$

$4 x$

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{3} - 6 x^{2} + 4 x$

$x^{2}$

$2 x$

$x^{2}$

$3 x$

$2$

$x^{4}$

$x^{3}$

$5 x^{2}$

$13 x$

$6$

$x^{4}$

$3 x^{3}$

$2 x^{2}$

$2 x^{3}$

$3 x^{2}$

$13 x$

$2 x^{3}$

$6 x^{2}$

$4 x$

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

$x^{2}$

$2 x$

$x^{2}$

$3 x$

$2$

$x^{4}$

$x^{3}$

$5 x^{2}$

$13 x$

$6$

$x^{4}$

$3 x^{3}$

$2 x^{2}$

$2 x^{3}$

$3 x^{2}$

$13 x$

$2 x^{3}$

$6 x^{2}$

$4 x$

$3 x^{2}$

$9 x$

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

$x^{2}$

$2 x$

$x^{2}$

$3 x$

$2$

$x^{4}$

$x^{3}$

$5 x^{2}$

$13 x$

$6$

$x^{4}$

$3 x^{3}$

$2 x^{2}$

$2 x^{3}$

$3 x^{2}$

$13 x$

$2 x^{3}$

$6 x^{2}$

$4 x$

$3 x^{2}$

$9 x$

$6$

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

$x^{2}$

$2 x$

$3$

$x^{2}$

$3 x$

$2$

$x^{4}$

$x^{3}$

$5 x^{2}$

$13 x$

$6$

$x^{4}$

$3 x^{3}$

$2 x^{2}$

$2 x^{3}$

$3 x^{2}$

$13 x$

$2 x^{3}$

$6 x^{2}$

$4 x$

$3 x^{2}$

$9 x$

$6$

Multiply the new quotient term by the divisor.

$x^{2}$

$2 x$

$3$

$x^{2}$

$3 x$

$2$

$x^{4}$

$x^{3}$

$5 x^{2}$

$13 x$

$6$

$x^{4}$

$3 x^{3}$

$2 x^{2}$

$2 x^{3}$

$3 x^{2}$

$13 x$

$2 x^{3}$

$6 x^{2}$

$4 x$

$3 x^{2}$

$9 x$

$6$

$3 x^{2}$

$9 x$

$6$

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{2} + 9 x - 6$

$x^{2}$

$2 x$

$3$

$x^{2}$

$3 x$

$2$

$x^{4}$

$x^{3}$

$5 x^{2}$

$13 x$

$6$

$x^{4}$

$3 x^{3}$

$2 x^{2}$

$2 x^{3}$

$3 x^{2}$

$13 x$

$2 x^{3}$

$6 x^{2}$

$4 x$

$3 x^{2}$

$9 x$

$6$

$3 x^{2}$

$9 x$

$6$

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

$x^{2}$

$2 x$

$3$

$x^{2}$

$3 x$

$2$

$x^{4}$

$x^{3}$

$5 x^{2}$

$13 x$

$6$

$x^{4}$

$3 x^{3}$

$2 x^{2}$

$2 x^{3}$

$3 x^{2}$

$13 x$

$2 x^{3}$

$6 x^{2}$

$4 x$

$3 x^{2}$

$9 x$

$6$

$3 x^{2}$

$9 x$

$6$

$0$

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

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

Divide Using Long Polynomial Division (x^4+x^3-5x^2+13x-6)/(x^2+3x-2)

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