Algebra Examples

$(10 x^{5} + x^{3} + 5 x^{2} - 2 x - 2) \div (5 x^{2} - 2)$

Step 1

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

$5 x^{2}$

$0 x$

$2$

$10 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$2 x$

$2$

Step 2

Divide the highest order term in the dividend $10 x^{5}$ by the highest order term in divisor $5 x^{2}$ .

$2 x^{3}$

$5 x^{2}$

$0 x$

$2$

$10 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$2 x$

$2$

Step 3

Multiply the new quotient term by the divisor.

$2 x^{3}$

$5 x^{2}$

$0 x$

$2$

$10 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$2 x$

$2$

$10 x^{5}$

$0$

$4 x^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{5} + 0 - 4 x^{3}$

$2 x^{3}$

$5 x^{2}$

$0 x$

$2$

$10 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$2 x$

$2$

$10 x^{5}$

$0$

$4 x^{3}$

Step 5

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

$2 x^{3}$

$5 x^{2}$

$0 x$

$2$

$10 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$2 x$

$2$

$10 x^{5}$

$0$

$4 x^{3}$

$5 x^{3}$

Step 6

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

$2 x^{3}$

$5 x^{2}$

$0 x$

$2$

$10 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$2 x$

$2$

$10 x^{5}$

$0$

$4 x^{3}$

$5 x^{3}$

$5 x^{2}$

$2 x$

Step 7

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

$2 x^{3}$

$0 x^{2}$

$x$

$5 x^{2}$

$0 x$

$2$

$10 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$2 x$

$2$

$10 x^{5}$

$0$

$4 x^{3}$

$5 x^{3}$

$5 x^{2}$

$2 x$

Step 8

Multiply the new quotient term by the divisor.

$2 x^{3}$

$0 x^{2}$

$x$

$5 x^{2}$

$0 x$

$2$

$10 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$2 x$

$2$

$10 x^{5}$

$0$

$4 x^{3}$

$5 x^{3}$

$5 x^{2}$

$2 x$

$5 x^{3}$

$0$

$2 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} + 0 - 2 x$

$2 x^{3}$

$0 x^{2}$

$x$

$5 x^{2}$

$0 x$

$2$

$10 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$2 x$

$2$

$10 x^{5}$

$0$

$4 x^{3}$

$5 x^{3}$

$5 x^{2}$

$2 x$

$5 x^{3}$

$0$

$2 x$

Step 10

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

$2 x^{3}$

$0 x^{2}$

$x$

$5 x^{2}$

$0 x$

$2$

$10 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$2 x$

$2$

$10 x^{5}$

$0$

$4 x^{3}$

$5 x^{3}$

$5 x^{2}$

$2 x$

$5 x^{3}$

$0$

$2 x$

$5 x^{2}$

$0$

Step 11

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

$2 x^{3}$

$0 x^{2}$

$x$

$5 x^{2}$

$0 x$

$2$

$10 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$2 x$

$2$

$10 x^{5}$

$0$

$4 x^{3}$

$5 x^{3}$

$5 x^{2}$

$2 x$

$5 x^{3}$

$0$

$2 x$

$5 x^{2}$

$0$

$2$

Step 12

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

$2 x^{3}$

$0 x^{2}$

$x$

$1$

$5 x^{2}$

$0 x$

$2$

$10 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$2 x$

$2$

$10 x^{5}$

$0$

$4 x^{3}$

$5 x^{3}$

$5 x^{2}$

$2 x$

$5 x^{3}$

$0$

$2 x$

$5 x^{2}$

$0$

$2$

Step 13

Multiply the new quotient term by the divisor.

$2 x^{3}$

$0 x^{2}$

$x$

$1$

$5 x^{2}$

$0 x$

$2$

$10 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$2 x$

$2$

$10 x^{5}$

$0$

$4 x^{3}$

$5 x^{3}$

$5 x^{2}$

$2 x$

$5 x^{3}$

$0$

$2 x$

$5 x^{2}$

$0$

$2$

$5 x^{2}$

$0$

$2$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{2} + 0 - 2$

$2 x^{3}$

$0 x^{2}$

$x$

$1$

$5 x^{2}$

$0 x$

$2$

$10 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$2 x$

$2$

$10 x^{5}$

$0$

$4 x^{3}$

$5 x^{3}$

$5 x^{2}$

$2 x$

$5 x^{3}$

$0$

$2 x$

$5 x^{2}$

$0$

$2$

$5 x^{2}$

$0$

$2$

Step 15

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

$2 x^{3}$

$0 x^{2}$

$x$

$1$

$5 x^{2}$

$0 x$

$2$

$10 x^{5}$

$0 x^{4}$

$x^{3}$

$5 x^{2}$

$2 x$

$2$

$10 x^{5}$

$0$

$4 x^{3}$

$5 x^{3}$

$5 x^{2}$

$2 x$

$5 x^{3}$

$0$

$2 x$

$5 x^{2}$

$0$

$2$

$5 x^{2}$

$0$

$2$

$0$

Step 16

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

$2 x^{3} + 0 x^{2} + x + 1$