Algebra Examples

$\frac{3 x^{5} + 17 x^{4} - 51 x^{2} + 3 x + 40}{x + 5}$

Step 1

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

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

Step 2

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

$3 x^{4}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

Step 3

Multiply the new quotient term by the divisor.

$3 x^{4}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{5} + 15 x^{4}$

$3 x^{4}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

Step 5

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

$3 x^{4}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

Step 6

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

$3 x^{4}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

Step 7

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

$3 x^{4}$

$2 x^{3}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

Step 8

Multiply the new quotient term by the divisor.

$3 x^{4}$

$2 x^{3}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{4} + 10 x^{3}$

$3 x^{4}$

$2 x^{3}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

Step 10

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

$3 x^{4}$

$2 x^{3}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

Step 11

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

$3 x^{4}$

$2 x^{3}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

$51 x^{2}$

Step 12

Divide the highest order term in the dividend $- 10 x^{3}$ by the highest order term in divisor $x$ .

$3 x^{4}$

$2 x^{3}$

$10 x^{2}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

$51 x^{2}$

Step 13

Multiply the new quotient term by the divisor.

$3 x^{4}$

$2 x^{3}$

$10 x^{2}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

$51 x^{2}$

$10 x^{3}$

$50 x^{2}$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{3} - 50 x^{2}$

$3 x^{4}$

$2 x^{3}$

$10 x^{2}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

$51 x^{2}$

$10 x^{3}$

$50 x^{2}$

Step 15

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

$3 x^{4}$

$2 x^{3}$

$10 x^{2}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

$51 x^{2}$

$10 x^{3}$

$50 x^{2}$

$x^{2}$

Step 16

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

$3 x^{4}$

$2 x^{3}$

$10 x^{2}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

$51 x^{2}$

$10 x^{3}$

$50 x^{2}$

$x^{2}$

$3 x$

Step 17

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

$3 x^{4}$

$2 x^{3}$

$10 x^{2}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

$51 x^{2}$

$10 x^{3}$

$50 x^{2}$

$x^{2}$

$3 x$

Step 18

Multiply the new quotient term by the divisor.

$3 x^{4}$

$2 x^{3}$

$10 x^{2}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

$51 x^{2}$

$10 x^{3}$

$50 x^{2}$

$x^{2}$

$3 x$

$x^{2}$

$5 x$

Step 19

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

$3 x^{4}$

$2 x^{3}$

$10 x^{2}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

$51 x^{2}$

$10 x^{3}$

$50 x^{2}$

$x^{2}$

$3 x$

$x^{2}$

$5 x$

Step 20

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

$3 x^{4}$

$2 x^{3}$

$10 x^{2}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

$51 x^{2}$

$10 x^{3}$

$50 x^{2}$

$x^{2}$

$3 x$

$x^{2}$

$5 x$

$8 x$

Step 21

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

$3 x^{4}$

$2 x^{3}$

$10 x^{2}$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

$51 x^{2}$

$10 x^{3}$

$50 x^{2}$

$x^{2}$

$3 x$

$x^{2}$

$5 x$

$8 x$

$40$

Step 22

Divide the highest order term in the dividend $8 x$ by the highest order term in divisor $x$ .

$3 x^{4}$

$2 x^{3}$

$10 x^{2}$

$x$

$8$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

$51 x^{2}$

$10 x^{3}$

$50 x^{2}$

$x^{2}$

$3 x$

$x^{2}$

$5 x$

$8 x$

$40$

Step 23

Multiply the new quotient term by the divisor.

$3 x^{4}$

$2 x^{3}$

$10 x^{2}$

$x$

$8$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

$51 x^{2}$

$10 x^{3}$

$50 x^{2}$

$x^{2}$

$3 x$

$x^{2}$

$5 x$

$8 x$

$40$

$8 x$

$40$

Step 24

The expression needs to be subtracted from the dividend, so change all the signs in $8 x + 40$

$3 x^{4}$

$2 x^{3}$

$10 x^{2}$

$x$

$8$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

$51 x^{2}$

$10 x^{3}$

$50 x^{2}$

$x^{2}$

$3 x$

$x^{2}$

$5 x$

$8 x$

$40$

$8 x$

$40$

Step 25

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

$3 x^{4}$

$2 x^{3}$

$10 x^{2}$

$x$

$8$

$x$

$5$

$3 x^{5}$

$17 x^{4}$

$0 x^{3}$

$51 x^{2}$

$3 x$

$40$

$3 x^{5}$

$15 x^{4}$

$2 x^{4}$

$0 x^{3}$

$2 x^{4}$

$10 x^{3}$

$51 x^{2}$

$10 x^{3}$

$50 x^{2}$

$x^{2}$

$3 x$

$x^{2}$

$5 x$

$8 x$

$40$

$8 x$

$40$

$0$

Step 26

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

$3 x^{4} + 2 x^{3} - 10 x^{2} - x + 8$