Algebra Examples

$\frac{(5 x^{5} + 90 x^{2} - 135 x)}{(x + 3)}$

Step 1

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

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

Step 2

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

$5 x^{4}$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

Step 3

Multiply the new quotient term by the divisor.

$5 x^{4}$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

Step 4

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

$5 x^{4}$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

Step 5

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

$5 x^{4}$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

Step 6

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

$5 x^{4}$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

$0 x^{3}$

Step 7

Divide the highest order term in the dividend $- 15 x^{4}$ by the highest order term in divisor $x$ .

$5 x^{4}$

$15 x^{3}$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

$0 x^{3}$

Step 8

Multiply the new quotient term by the divisor.

$5 x^{4}$

$15 x^{3}$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

$0 x^{3}$

$15 x^{4}$

$45 x^{3}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 15 x^{4} - 45 x^{3}$

$5 x^{4}$

$15 x^{3}$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

$0 x^{3}$

$15 x^{4}$

$45 x^{3}$

Step 10

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

$5 x^{4}$

$15 x^{3}$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

$0 x^{3}$

$15 x^{4}$

$45 x^{3}$

Step 11

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

$5 x^{4}$

$15 x^{3}$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

$0 x^{3}$

$15 x^{4}$

$45 x^{3}$

$90 x^{2}$

Step 12

Divide the highest order term in the dividend $45 x^{3}$ by the highest order term in divisor $x$ .

$5 x^{4}$

$15 x^{3}$

$45 x^{2}$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

$0 x^{3}$

$15 x^{4}$

$45 x^{3}$

$90 x^{2}$

Step 13

Multiply the new quotient term by the divisor.

$5 x^{4}$

$15 x^{3}$

$45 x^{2}$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

$0 x^{3}$

$15 x^{4}$

$45 x^{3}$

$90 x^{2}$

$45 x^{3}$

$135 x^{2}$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $45 x^{3} + 135 x^{2}$

$5 x^{4}$

$15 x^{3}$

$45 x^{2}$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

$0 x^{3}$

$15 x^{4}$

$45 x^{3}$

$90 x^{2}$

$45 x^{3}$

$135 x^{2}$

Step 15

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

$5 x^{4}$

$15 x^{3}$

$45 x^{2}$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

$0 x^{3}$

$15 x^{4}$

$45 x^{3}$

$90 x^{2}$

$45 x^{3}$

$135 x^{2}$

$45 x^{2}$

Step 16

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

$5 x^{4}$

$15 x^{3}$

$45 x^{2}$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

$0 x^{3}$

$15 x^{4}$

$45 x^{3}$

$90 x^{2}$

$45 x^{3}$

$135 x^{2}$

$45 x^{2}$

$135 x$

Step 17

Divide the highest order term in the dividend $- 45 x^{2}$ by the highest order term in divisor $x$ .

$5 x^{4}$

$15 x^{3}$

$45 x^{2}$

$45 x$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

$0 x^{3}$

$15 x^{4}$

$45 x^{3}$

$90 x^{2}$

$45 x^{3}$

$135 x^{2}$

$45 x^{2}$

$135 x$

Step 18

Multiply the new quotient term by the divisor.

$5 x^{4}$

$15 x^{3}$

$45 x^{2}$

$45 x$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

$0 x^{3}$

$15 x^{4}$

$45 x^{3}$

$90 x^{2}$

$45 x^{3}$

$135 x^{2}$

$45 x^{2}$

$135 x$

$45 x^{2}$

$135 x$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $- 45 x^{2} - 135 x$

$5 x^{4}$

$15 x^{3}$

$45 x^{2}$

$45 x$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

$0 x^{3}$

$15 x^{4}$

$45 x^{3}$

$90 x^{2}$

$45 x^{3}$

$135 x^{2}$

$45 x^{2}$

$135 x$

$45 x^{2}$

$135 x$

Step 20

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

$5 x^{4}$

$15 x^{3}$

$45 x^{2}$

$45 x$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

$0 x^{3}$

$15 x^{4}$

$45 x^{3}$

$90 x^{2}$

$45 x^{3}$

$135 x^{2}$

$45 x^{2}$

$135 x$

$45 x^{2}$

$135 x$

$0$

Step 21

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

$5 x^{4}$

$15 x^{3}$

$45 x^{2}$

$45 x$

$x$

$3$

$5 x^{5}$

$0 x^{4}$

$0 x^{3}$

$90 x^{2}$

$135 x$

$0$

$5 x^{5}$

$15 x^{4}$

$0 x^{3}$

$15 x^{4}$

$45 x^{3}$

$90 x^{2}$

$45 x^{3}$

$135 x^{2}$

$45 x^{2}$

$135 x$

$45 x^{2}$

$135 x$

$0$

Step 22

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

$5 x^{4} - 15 x^{3} + 45 x^{2} - 45 x$