Algebra Examples

$(8 x^{4} - 12 x + 1) \div (2 x^{2} - 1)$

Step 1

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

$2 x^{2}$

$0 x$

$1$

$8 x^{4}$

$0 x^{3}$

$0 x^{2}$

$12 x$

$1$

Step 2

Divide the highest order term in the dividend $8 x^{4}$ by the highest order term in divisor $2 x^{2}$ .

$4 x^{2}$

$2 x^{2}$

$0 x$

$1$

$8 x^{4}$

$0 x^{3}$

$0 x^{2}$

$12 x$

$1$

Step 3

Multiply the new quotient term by the divisor.

$4 x^{2}$

$2 x^{2}$

$0 x$

$1$

$8 x^{4}$

$0 x^{3}$

$0 x^{2}$

$12 x$

$1$

$8 x^{4}$

$0$

$4 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{4} + 0 - 4 x^{2}$

$4 x^{2}$

$2 x^{2}$

$0 x$

$1$

$8 x^{4}$

$0 x^{3}$

$0 x^{2}$

$12 x$

$1$

$8 x^{4}$

$0$

$4 x^{2}$

Step 5

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

$4 x^{2}$

$2 x^{2}$

$0 x$

$1$

$8 x^{4}$

$0 x^{3}$

$0 x^{2}$

$12 x$

$1$

$8 x^{4}$

$0$

$4 x^{2}$

Step 6

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

$4 x^{2}$

$2 x^{2}$

$0 x$

$1$

$8 x^{4}$

$0 x^{3}$

$0 x^{2}$

$12 x$

$1$

$8 x^{4}$

$0$

$4 x^{2}$

$12 x$

$1$

Step 7

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

$4 x^{2}$

$0 x$

$2$

$2 x^{2}$

$0 x$

$1$

$8 x^{4}$

$0 x^{3}$

$0 x^{2}$

$12 x$

$1$

$8 x^{4}$

$0$

$4 x^{2}$

$12 x$

$1$

Step 8

Multiply the new quotient term by the divisor.

$4 x^{2}$

$0 x$

$2$

$2 x^{2}$

$0 x$

$1$

$8 x^{4}$

$0 x^{3}$

$0 x^{2}$

$12 x$

$1$

$8 x^{4}$

$0$

$4 x^{2}$

$12 x$

$1$

$4 x^{2}$

$0$

$2$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{2} + 0 - 2$

$4 x^{2}$

$0 x$

$2$

$2 x^{2}$

$0 x$

$1$

$8 x^{4}$

$0 x^{3}$

$0 x^{2}$

$12 x$

$1$

$8 x^{4}$

$0$

$4 x^{2}$

$12 x$

$1$

$4 x^{2}$

$0$

$2$

Step 10

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

$4 x^{2}$

$0 x$

$2$

$2 x^{2}$

$0 x$

$1$

$8 x^{4}$

$0 x^{3}$

$0 x^{2}$

$12 x$

$1$

$8 x^{4}$

$0$

$4 x^{2}$

$12 x$

$1$

$4 x^{2}$

$0$

$2$

$12 x$

$3$

Step 11

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

$4 x^{2} + 2$