Algebra Examples

$(2 x^{3} + x^{4} - 6 x^{2} + 11 x - 10) \div (x^{2} + 2 - x)$

Step 1

Reorder $2 x^{3}$ and $x^{4}$ .

$\frac{x^{4} + 2 x^{3} - 6 x^{2} + 11 x - 10}{x^{2} + 2 - x}$

Step 2

Move $2$ .

$\frac{x^{4} + 2 x^{3} - 6 x^{2} + 11 x - 10}{x^{2} - x + 2}$

Step 3

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

$x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$11 x$

$10$

Step 4

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

$x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$11 x$

$10$

Step 5

Multiply the new quotient term by the divisor.

$x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$11 x$

$10$

$x^{4}$

$x^{3}$

$2 x^{2}$

Step 6

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

$x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$11 x$

$10$

$x^{4}$

$x^{3}$

$2 x^{2}$

Step 7

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

$x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$11 x$

$10$

$x^{4}$

$x^{3}$

$2 x^{2}$

$3 x^{3}$

$8 x^{2}$

Step 8

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

$x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$11 x$

$10$

$x^{4}$

$x^{3}$

$2 x^{2}$

$3 x^{3}$

$8 x^{2}$

$11 x$

Step 9

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

$x^{2}$

$3 x$

$x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$11 x$

$10$

$x^{4}$

$x^{3}$

$2 x^{2}$

$3 x^{3}$

$8 x^{2}$

$11 x$

Step 10

Multiply the new quotient term by the divisor.

$x^{2}$

$3 x$

$x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$11 x$

$10$

$x^{4}$

$x^{3}$

$2 x^{2}$

$3 x^{3}$

$8 x^{2}$

$11 x$

$3 x^{3}$

$3 x^{2}$

$6 x$

Step 11

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

$x^{2}$

$3 x$

$x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$11 x$

$10$

$x^{4}$

$x^{3}$

$2 x^{2}$

$3 x^{3}$

$8 x^{2}$

$11 x$

$3 x^{3}$

$3 x^{2}$

$6 x$

Step 12

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

$x^{2}$

$3 x$

$x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$11 x$

$10$

$x^{4}$

$x^{3}$

$2 x^{2}$

$3 x^{3}$

$8 x^{2}$

$11 x$

$3 x^{3}$

$3 x^{2}$

$6 x$

$5 x^{2}$

$5 x$

Step 13

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

$x^{2}$

$3 x$

$x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$11 x$

$10$

$x^{4}$

$x^{3}$

$2 x^{2}$

$3 x^{3}$

$8 x^{2}$

$11 x$

$3 x^{3}$

$3 x^{2}$

$6 x$

$5 x^{2}$

$5 x$

$10$

Step 14

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

$x^{2}$

$3 x$

$5$

$x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$11 x$

$10$

$x^{4}$

$x^{3}$

$2 x^{2}$

$3 x^{3}$

$8 x^{2}$

$11 x$

$3 x^{3}$

$3 x^{2}$

$6 x$

$5 x^{2}$

$5 x$

$10$

Step 15

Multiply the new quotient term by the divisor.

$x^{2}$

$3 x$

$5$

$x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$11 x$

$10$

$x^{4}$

$x^{3}$

$2 x^{2}$

$3 x^{3}$

$8 x^{2}$

$11 x$

$3 x^{3}$

$3 x^{2}$

$6 x$

$5 x^{2}$

$5 x$

$10$

$5 x^{2}$

$5 x$

$10$

Step 16

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x^{2} + 5 x - 10$

$x^{2}$

$3 x$

$5$

$x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$11 x$

$10$

$x^{4}$

$x^{3}$

$2 x^{2}$

$3 x^{3}$

$8 x^{2}$

$11 x$

$3 x^{3}$

$3 x^{2}$

$6 x$

$5 x^{2}$

$5 x$

$10$

$5 x^{2}$

$5 x$

$10$

Step 17

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

$x^{2}$

$3 x$

$5$

$x^{2}$

$x$

$2$

$x^{4}$

$2 x^{3}$

$6 x^{2}$

$11 x$

$10$

$x^{4}$

$x^{3}$

$2 x^{2}$

$3 x^{3}$

$8 x^{2}$

$11 x$

$3 x^{3}$

$3 x^{2}$

$6 x$

$5 x^{2}$

$5 x$

$10$

$5 x^{2}$

$5 x$

$10$

$0$

Step 18

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

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