Algebra Examples

$(- 19 x^{2} + 3 + 7 x + 15 x^{4}) \div (- 5 x^{2} + 3)$

Step 1

To calculate the remainder, first divide the polynomials.

Tap for more steps...

Step 1.1

Expand $- 19 x^{2} + 3 + 7 x + 15 x^{4}$ .

Tap for more steps...

Step 1.1.1

Move $3$ .

$\frac{- 19 x^{2} + 7 x + 3 + 15 x^{4}}{- 5 x^{2} + 3}$

Step 1.1.2

Move $3$ .

$\frac{- 19 x^{2} + 7 x + 15 x^{4} + 3}{- 5 x^{2} + 3}$

Step 1.1.3

Move $7 x$ .

$\frac{- 19 x^{2} + 15 x^{4} + 7 x + 3}{- 5 x^{2} + 3}$

Step 1.1.4

Reorder $- 19 x^{2}$ and $15 x^{4}$ .

$\frac{15 x^{4} - 19 x^{2} + 7 x + 3}{- 5 x^{2} + 3}$

Step 1.2

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

$5 x^{2}$

$0 x$

$3$

$15 x^{4}$

$0 x^{3}$

$19 x^{2}$

$7 x$

$3$

Step 1.3

Divide the highest order term in the dividend $15 x^{4}$ by the highest order term in divisor $- 5 x^{2}$ .

$3 x^{2}$

$5 x^{2}$

$0 x$

$3$

$15 x^{4}$

$0 x^{3}$

$19 x^{2}$

$7 x$

$3$

Step 1.4

Multiply the new quotient term by the divisor.

$3 x^{2}$

$5 x^{2}$

$0 x$

$3$

$15 x^{4}$

$0 x^{3}$

$19 x^{2}$

$7 x$

$3$

$15 x^{4}$

$0$

$9 x^{2}$

Step 1.5

The expression needs to be subtracted from the dividend, so change all the signs in $15 x^{4} + 0 - 9 x^{2}$

$3 x^{2}$

$5 x^{2}$

$0 x$

$3$

$15 x^{4}$

$0 x^{3}$

$19 x^{2}$

$7 x$

$3$

$15 x^{4}$

$0$

$9 x^{2}$

Step 1.6

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

$3 x^{2}$

$5 x^{2}$

$0 x$

$3$

$15 x^{4}$

$0 x^{3}$

$19 x^{2}$

$7 x$

$3$

$15 x^{4}$

$0$

$9 x^{2}$

$10 x^{2}$

Step 1.7

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

$3 x^{2}$

$5 x^{2}$

$0 x$

$3$

$15 x^{4}$

$0 x^{3}$

$19 x^{2}$

$7 x$

$3$

$15 x^{4}$

$0$

$9 x^{2}$

$10 x^{2}$

$7 x$

$3$

Step 1.8

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

$3 x^{2}$

$0 x$

$2$

$5 x^{2}$

$0 x$

$3$

$15 x^{4}$

$0 x^{3}$

$19 x^{2}$

$7 x$

$3$

$15 x^{4}$

$0$

$9 x^{2}$

$10 x^{2}$

$7 x$

$3$

Step 1.9

Multiply the new quotient term by the divisor.

$3 x^{2}$

$0 x$

$2$

$5 x^{2}$

$0 x$

$3$

$15 x^{4}$

$0 x^{3}$

$19 x^{2}$

$7 x$

$3$

$15 x^{4}$

$0$

$9 x^{2}$

$10 x^{2}$

$7 x$

$3$

$10 x^{2}$

$0$

$6$

Step 1.10

The expression needs to be subtracted from the dividend, so change all the signs in $- 10 x^{2} + 0 + 6$

$3 x^{2}$

$0 x$

$2$

$5 x^{2}$

$0 x$

$3$

$15 x^{4}$

$0 x^{3}$

$19 x^{2}$

$7 x$

$3$

$15 x^{4}$

$0$

$9 x^{2}$

$10 x^{2}$

$7 x$

$3$

$10 x^{2}$

$0$

$6$

Step 1.11

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

$3 x^{2}$

$0 x$

$2$

$5 x^{2}$

$0 x$

$3$

$15 x^{4}$

$0 x^{3}$

$19 x^{2}$

$7 x$

$3$

$15 x^{4}$

$0$

$9 x^{2}$

$10 x^{2}$

$7 x$

$3$

$10 x^{2}$

$0$

$6$

$7 x$

$3$

Step 1.12

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

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

Step 2

Since the last term in the resulting expression is a fraction, the numerator of the fraction is the remainder.

$7 x - 3$