Algebra Examples

$\frac{3 x^{5} - x^{3} + 3 x^{2} - 133 x - 21}{x^{2} - 7}$

Step 1

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

$x^{2}$

$0 x$

$7$

$3 x^{5}$

$0 x^{4}$

$x^{3}$

$3 x^{2}$

$133 x$

$21$

Step 2

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

$3 x^{3}$

$x^{2}$

$0 x$

$7$

$3 x^{5}$

$0 x^{4}$

$x^{3}$

$3 x^{2}$

$133 x$

$21$

Step 3

Multiply the new quotient term by the divisor.

$3 x^{3}$

$x^{2}$

$0 x$

$7$

$3 x^{5}$

$0 x^{4}$

$x^{3}$

$3 x^{2}$

$133 x$

$21$

$3 x^{5}$

$0$

$21 x^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{5} + 0 - 21 x^{3}$

$3 x^{3}$

$x^{2}$

$0 x$

$7$

$3 x^{5}$

$0 x^{4}$

$x^{3}$

$3 x^{2}$

$133 x$

$21$

$3 x^{5}$

$0$

$21 x^{3}$

Step 5

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

$3 x^{3}$

$x^{2}$

$0 x$

$7$

$3 x^{5}$

$0 x^{4}$

$x^{3}$

$3 x^{2}$

$133 x$

$21$

$3 x^{5}$

$0$

$21 x^{3}$

$20 x^{3}$

Step 6

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

$3 x^{3}$

$x^{2}$

$0 x$

$7$

$3 x^{5}$

$0 x^{4}$

$x^{3}$

$3 x^{2}$

$133 x$

$21$

$3 x^{5}$

$0$

$21 x^{3}$

$20 x^{3}$

$3 x^{2}$

$133 x$

Step 7

Divide the highest order term in the dividend $20 x^{3}$ by the highest order term in divisor $x^{2}$ .

$3 x^{3}$

$0 x^{2}$

$20 x$

$x^{2}$

$0 x$

$7$

$3 x^{5}$

$0 x^{4}$

$x^{3}$

$3 x^{2}$

$133 x$

$21$

$3 x^{5}$

$0$

$21 x^{3}$

$20 x^{3}$

$3 x^{2}$

$133 x$

Step 8

Multiply the new quotient term by the divisor.

$3 x^{3}$

$0 x^{2}$

$20 x$

$x^{2}$

$0 x$

$7$

$3 x^{5}$

$0 x^{4}$

$x^{3}$

$3 x^{2}$

$133 x$

$21$

$3 x^{5}$

$0$

$21 x^{3}$

$20 x^{3}$

$3 x^{2}$

$133 x$

$20 x^{3}$

$0$

$140 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $20 x^{3} + 0 - 140 x$

$3 x^{3}$

$0 x^{2}$

$20 x$

$x^{2}$

$0 x$

$7$

$3 x^{5}$

$0 x^{4}$

$x^{3}$

$3 x^{2}$

$133 x$

$21$

$3 x^{5}$

$0$

$21 x^{3}$

$20 x^{3}$

$3 x^{2}$

$133 x$

$20 x^{3}$

$0$

$140 x$

Step 10

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

$3 x^{3}$

$0 x^{2}$

$20 x$

$x^{2}$

$0 x$

$7$

$3 x^{5}$

$0 x^{4}$

$x^{3}$

$3 x^{2}$

$133 x$

$21$

$3 x^{5}$

$0$

$21 x^{3}$

$20 x^{3}$

$3 x^{2}$

$133 x$

$20 x^{3}$

$0$

$140 x$

$3 x^{2}$

$7 x$

Step 11

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

$3 x^{3}$

$0 x^{2}$

$20 x$

$x^{2}$

$0 x$

$7$

$3 x^{5}$

$0 x^{4}$

$x^{3}$

$3 x^{2}$

$133 x$

$21$

$3 x^{5}$

$0$

$21 x^{3}$

$20 x^{3}$

$3 x^{2}$

$133 x$

$20 x^{3}$

$0$

$140 x$

$3 x^{2}$

$7 x$

$21$

Step 12

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

$3 x^{3}$

$0 x^{2}$

$20 x$

$3$

$x^{2}$

$0 x$

$7$

$3 x^{5}$

$0 x^{4}$

$x^{3}$

$3 x^{2}$

$133 x$

$21$

$3 x^{5}$

$0$

$21 x^{3}$

$20 x^{3}$

$3 x^{2}$

$133 x$

$20 x^{3}$

$0$

$140 x$

$3 x^{2}$

$7 x$

$21$

Step 13

Multiply the new quotient term by the divisor.

$3 x^{3}$

$0 x^{2}$

$20 x$

$3$

$x^{2}$

$0 x$

$7$

$3 x^{5}$

$0 x^{4}$

$x^{3}$

$3 x^{2}$

$133 x$

$21$

$3 x^{5}$

$0$

$21 x^{3}$

$20 x^{3}$

$3 x^{2}$

$133 x$

$20 x^{3}$

$0$

$140 x$

$3 x^{2}$

$7 x$

$21$

$3 x^{2}$

$0$

$21$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{2} + 0 - 21$

$3 x^{3}$

$0 x^{2}$

$20 x$

$3$

$x^{2}$

$0 x$

$7$

$3 x^{5}$

$0 x^{4}$

$x^{3}$

$3 x^{2}$

$133 x$

$21$

$3 x^{5}$

$0$

$21 x^{3}$

$20 x^{3}$

$3 x^{2}$

$133 x$

$20 x^{3}$

$0$

$140 x$

$3 x^{2}$

$7 x$

$21$

$3 x^{2}$

$0$

$21$

Step 15

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

$3 x^{3}$

$0 x^{2}$

$20 x$

$3$

$x^{2}$

$0 x$

$7$

$3 x^{5}$

$0 x^{4}$

$x^{3}$

$3 x^{2}$

$133 x$

$21$

$3 x^{5}$

$0$

$21 x^{3}$

$20 x^{3}$

$3 x^{2}$

$133 x$

$20 x^{3}$

$0$

$140 x$

$3 x^{2}$

$7 x$

$21$

$3 x^{2}$

$0$

$21$

$7 x$

$0$

Step 16

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

$3 x^{3} + 20 x + 3 + \frac{7 x}{x^{2} - 7}$