Algebra Examples

$\frac{x^{5} + 2 x^{3} - x - 8}{x^{2} - 2 x + 1}$

Step 1

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

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

Step 2

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

$x^{3}$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

Step 3

Multiply the new quotient term by the divisor.

$x^{3}$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $x^{5} - 2 x^{4} + x^{3}$

$x^{3}$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

Step 5

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

$x^{3}$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$x^{3}$

Step 6

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

$x^{3}$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

Step 7

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

$x^{3}$

$2 x^{2}$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x^{2}$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

Step 9

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

$x^{3}$

$2 x^{2}$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

Step 10

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

$x^{3}$

$2 x^{2}$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$5 x^{3}$

$2 x^{2}$

Step 11

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

$x^{3}$

$2 x^{2}$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$5 x^{3}$

$2 x^{2}$

$x$

Step 12

Divide the highest order term in the dividend $5 x^{3}$ by the highest order term in divisor $x^{2}$ .

$x^{3}$

$2 x^{2}$

$5 x$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$5 x^{3}$

$2 x^{2}$

$x$

Step 13

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x^{2}$

$5 x$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$5 x^{3}$

$2 x^{2}$

$x$

$5 x^{3}$

$10 x^{2}$

$5 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} - 10 x^{2} + 5 x$

$x^{3}$

$2 x^{2}$

$5 x$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$5 x^{3}$

$2 x^{2}$

$x$

$5 x^{3}$

$10 x^{2}$

$5 x$

Step 15

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

$x^{3}$

$2 x^{2}$

$5 x$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$5 x^{3}$

$2 x^{2}$

$x$

$5 x^{3}$

$10 x^{2}$

$5 x$

$8 x^{2}$

$6 x$

Step 16

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

$x^{3}$

$2 x^{2}$

$5 x$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$5 x^{3}$

$2 x^{2}$

$x$

$5 x^{3}$

$10 x^{2}$

$5 x$

$8 x^{2}$

$6 x$

$8$

Step 17

Divide the highest order term in the dividend $8 x^{2}$ by the highest order term in divisor $x^{2}$ .

$x^{3}$

$2 x^{2}$

$5 x$

$8$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$5 x^{3}$

$2 x^{2}$

$x$

$5 x^{3}$

$10 x^{2}$

$5 x$

$8 x^{2}$

$6 x$

$8$

Step 18

Multiply the new quotient term by the divisor.

$x^{3}$

$2 x^{2}$

$5 x$

$8$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$5 x^{3}$

$2 x^{2}$

$x$

$5 x^{3}$

$10 x^{2}$

$5 x$

$8 x^{2}$

$6 x$

$8$

$8 x^{2}$

$16 x$

$8$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{2} - 16 x + 8$

$x^{3}$

$2 x^{2}$

$5 x$

$8$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$5 x^{3}$

$2 x^{2}$

$x$

$5 x^{3}$

$10 x^{2}$

$5 x$

$8 x^{2}$

$6 x$

$8$

$8 x^{2}$

$16 x$

$8$

Step 20

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

$x^{3}$

$2 x^{2}$

$5 x$

$8$

$x^{2}$

$2 x$

$1$

$x^{5}$

$0 x^{4}$

$2 x^{3}$

$0 x^{2}$

$x$

$8$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$x^{3}$

$0 x^{2}$

$2 x^{4}$

$4 x^{3}$

$2 x^{2}$

$5 x^{3}$

$2 x^{2}$

$x$

$5 x^{3}$

$10 x^{2}$

$5 x$

$8 x^{2}$

$6 x$

$8$

$8 x^{2}$

$16 x$

$8$

$10 x$

$16$

Step 21

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

$x^{3} + 2 x^{2} + 5 x + 8 + \frac{10 x - 16}{x^{2} - 2 x + 1}$