Divide Using Long Polynomial Division (x^4+x^3+5x^2-10x+10)/(x^2+2x+1)

$\frac{x^{4} + x^{3} + 5 x^{2} - 10 x + 10}{x^{2} + 2 x + 1}$

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

$x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$10 x$

$10$

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

$x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$10 x$

$10$

Multiply the new quotient term by the divisor.

$x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$10 x$

$10$

$x^{4}$

$2 x^{3}$

$x^{2}$

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

$x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$10 x$

$10$

$x^{4}$

$2 x^{3}$

$x^{2}$

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

$x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$10 x$

$10$

$x^{4}$

$2 x^{3}$

$x^{2}$

$x^{3}$

$4 x^{2}$

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

$x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$10 x$

$10$

$x^{4}$

$2 x^{3}$

$x^{2}$

$x^{3}$

$4 x^{2}$

$10 x$

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

$x^{2}$

$x$

$x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$10 x$

$10$

$x^{4}$

$2 x^{3}$

$x^{2}$

$x^{3}$

$4 x^{2}$

$10 x$

Multiply the new quotient term by the divisor.

$x^{2}$

$x$

$x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$10 x$

$10$

$x^{4}$

$2 x^{3}$

$x^{2}$

$x^{3}$

$4 x^{2}$

$10 x$

$x^{3}$

$2 x^{2}$

$x$

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

$x^{2}$

$x$

$x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$10 x$

$10$

$x^{4}$

$2 x^{3}$

$x^{2}$

$x^{3}$

$4 x^{2}$

$10 x$

$x^{3}$

$2 x^{2}$

$x$

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

$x^{2}$

$x$

$x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$10 x$

$10$

$x^{4}$

$2 x^{3}$

$x^{2}$

$x^{3}$

$4 x^{2}$

$10 x$

$x^{3}$

$2 x^{2}$

$x$

$6 x^{2}$

$9 x$

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

$x^{2}$

$x$

$x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$10 x$

$10$

$x^{4}$

$2 x^{3}$

$x^{2}$

$x^{3}$

$4 x^{2}$

$10 x$

$x^{3}$

$2 x^{2}$

$x$

$6 x^{2}$

$9 x$

$10$

Divide the highest order term in the dividend $6 x^{2}$ by the highest order term in divisor $x^{2}$ .

$x^{2}$

$x$

$6$

$x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$10 x$

$10$

$x^{4}$

$2 x^{3}$

$x^{2}$

$x^{3}$

$4 x^{2}$

$10 x$

$x^{3}$

$2 x^{2}$

$x$

$6 x^{2}$

$9 x$

$10$

Multiply the new quotient term by the divisor.

$x^{2}$

$x$

$6$

$x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$10 x$

$10$

$x^{4}$

$2 x^{3}$

$x^{2}$

$x^{3}$

$4 x^{2}$

$10 x$

$x^{3}$

$2 x^{2}$

$x$

$6 x^{2}$

$9 x$

$10$

$6 x^{2}$

$12 x$

$6$

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} + 12 x + 6$

$x^{2}$

$x$

$6$

$x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$10 x$

$10$

$x^{4}$

$2 x^{3}$

$x^{2}$

$x^{3}$

$4 x^{2}$

$10 x$

$x^{3}$

$2 x^{2}$

$x$

$6 x^{2}$

$9 x$

$10$

$6 x^{2}$

$12 x$

$6$

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

$x^{2}$

$x$

$6$

$x^{2}$

$2 x$

$1$

$x^{4}$

$x^{3}$

$5 x^{2}$

$10 x$

$10$

$x^{4}$

$2 x^{3}$

$x^{2}$

$x^{3}$

$4 x^{2}$

$10 x$

$x^{3}$

$2 x^{2}$

$x$

$6 x^{2}$

$9 x$

$10$

$6 x^{2}$

$12 x$

$6$

$21 x$

$4$

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

$x^{2} - x + 6 + \frac{- 21 x + 4}{x^{2} + 2 x + 1}$

Divide Using Long Polynomial Division (x^4+x^3+5x^2-10x+10)/(x^2+2x+1)

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