Algebra Examples

$(x^{5} - x^{3} + x - 5) \div (x - 2)$

Step 1

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

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

Step 2

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

$x^{4}$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

Step 3

Multiply the new quotient term by the divisor.

$x^{4}$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

Step 4

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

$x^{4}$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

Step 5

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

$x^{4}$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

Step 6

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

$x^{4}$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

Step 7

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

$x^{4}$

$2 x^{3}$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

Step 8

Multiply the new quotient term by the divisor.

$x^{4}$

$2 x^{3}$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

Step 9

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

$x^{4}$

$2 x^{3}$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

Step 10

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

$x^{4}$

$2 x^{3}$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

$3 x^{3}$

Step 11

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

$x^{4}$

$2 x^{3}$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

$3 x^{3}$

$0 x^{2}$

Step 12

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

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

$3 x^{3}$

$0 x^{2}$

Step 13

Multiply the new quotient term by the divisor.

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$6 x^{2}$

Step 14

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

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$6 x^{2}$

Step 15

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

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$6 x^{2}$

Step 16

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

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$6 x^{2}$

$x$

Step 17

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

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$6 x$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$6 x^{2}$

$x$

Step 18

Multiply the new quotient term by the divisor.

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$6 x$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$6 x^{2}$

$x$

$6 x^{2}$

$12 x$

Step 19

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

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$6 x$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$6 x^{2}$

$x$

$6 x^{2}$

$12 x$

Step 20

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

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$6 x$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$6 x^{2}$

$x$

$6 x^{2}$

$12 x$

$13 x$

Step 21

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

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$6 x$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$6 x^{2}$

$x$

$6 x^{2}$

$12 x$

$13 x$

$5$

Step 22

Divide the highest order term in the dividend $13 x$ by the highest order term in divisor $x$ .

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$6 x$

$13$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$6 x^{2}$

$x$

$6 x^{2}$

$12 x$

$13 x$

$5$

Step 23

Multiply the new quotient term by the divisor.

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$6 x$

$13$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$6 x^{2}$

$x$

$6 x^{2}$

$12 x$

$13 x$

$5$

$13 x$

$26$

Step 24

The expression needs to be subtracted from the dividend, so change all the signs in $13 x - 26$

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$6 x$

$13$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$6 x^{2}$

$x$

$6 x^{2}$

$12 x$

$13 x$

$5$

$13 x$

$26$

Step 25

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

$x^{4}$

$2 x^{3}$

$3 x^{2}$

$6 x$

$13$

$x$

$2$

$x^{5}$

$0 x^{4}$

$x^{3}$

$0 x^{2}$

$x$

$5$

$x^{5}$

$2 x^{4}$

$x^{3}$

$2 x^{4}$

$4 x^{3}$

$3 x^{3}$

$0 x^{2}$

$3 x^{3}$

$6 x^{2}$

$x$

$6 x^{2}$

$12 x$

$13 x$

$5$

$13 x$

$26$

$21$

Step 26

The quotient is $x^{4} + 2 x^{3} + 3 x^{2} + 6 x + 13$ .

$x^{4} + 2 x^{3} + 3 x^{2} + 6 x + 13$