Algebra Examples

$(10 x^{3} + 3 x^{2} + 5 x - 2) \div (2 x^{2} - 3 x - 1)$

Step 1

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

$2 x^{2}$

$3 x$

$1$

$10 x^{3}$

$3 x^{2}$

$5 x$

$2$

Step 2

Divide the highest order term in the dividend $10 x^{3}$ by the highest order term in divisor $2 x^{2}$ .

$5 x$

$2 x^{2}$

$3 x$

$1$

$10 x^{3}$

$3 x^{2}$

$5 x$

$2$

Step 3

Multiply the new quotient term by the divisor.

$5 x$

$2 x^{2}$

$3 x$

$1$

$10 x^{3}$

$3 x^{2}$

$5 x$

$2$

$10 x^{3}$

$15 x^{2}$

$5 x$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $10 x^{3} - 15 x^{2} - 5 x$

$5 x$

$2 x^{2}$

$3 x$

$1$

$10 x^{3}$

$3 x^{2}$

$5 x$

$2$

$10 x^{3}$

$15 x^{2}$

$5 x$

Step 5

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

$5 x$

$2 x^{2}$

$3 x$

$1$

$10 x^{3}$

$3 x^{2}$

$5 x$

$2$

$10 x^{3}$

$15 x^{2}$

$5 x$

$18 x^{2}$

$10 x$

Step 6

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

$5 x$

$2 x^{2}$

$3 x$

$1$

$10 x^{3}$

$3 x^{2}$

$5 x$

$2$

$10 x^{3}$

$15 x^{2}$

$5 x$

$18 x^{2}$

$10 x$

$2$

Step 7

Divide the highest order term in the dividend $18 x^{2}$ by the highest order term in divisor $2 x^{2}$ .

$5 x$

$9$

$2 x^{2}$

$3 x$

$1$

$10 x^{3}$

$3 x^{2}$

$5 x$

$2$

$10 x^{3}$

$15 x^{2}$

$5 x$

$18 x^{2}$

$10 x$

$2$

Step 8

Multiply the new quotient term by the divisor.

$5 x$

$9$

$2 x^{2}$

$3 x$

$1$

$10 x^{3}$

$3 x^{2}$

$5 x$

$2$

$10 x^{3}$

$15 x^{2}$

$5 x$

$18 x^{2}$

$10 x$

$2$

$18 x^{2}$

$27 x$

$9$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $18 x^{2} - 27 x - 9$

$5 x$

$9$

$2 x^{2}$

$3 x$

$1$

$10 x^{3}$

$3 x^{2}$

$5 x$

$2$

$10 x^{3}$

$15 x^{2}$

$5 x$

$18 x^{2}$

$10 x$

$2$

$18 x^{2}$

$27 x$

$9$

Step 10

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

$5 x$

$9$

$2 x^{2}$

$3 x$

$1$

$10 x^{3}$

$3 x^{2}$

$5 x$

$2$

$10 x^{3}$

$15 x^{2}$

$5 x$

$18 x^{2}$

$10 x$

$2$

$18 x^{2}$

$27 x$

$9$

$37 x$

$7$

Step 11

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

$5 x + 9 + \frac{37 x + 7}{2 x^{2} - 3 x - 1}$