Algebra Examples

$(- 7 x^{4} + 5 x^{3} + 20 x^{2} + 10) \div (- x^{2} + x + 3)$

Step 1

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

$x^{2}$

$x$

$3$

$7 x^{4}$

$5 x^{3}$

$20 x^{2}$

$0 x$

$10$

Step 2

Divide the highest order term in the dividend $- 7 x^{4}$ by the highest order term in divisor $- x^{2}$ .

$7 x^{2}$

$x^{2}$

$x$

$3$

$7 x^{4}$

$5 x^{3}$

$20 x^{2}$

$0 x$

$10$

Step 3

Multiply the new quotient term by the divisor.

$7 x^{2}$

$x^{2}$

$x$

$3$

$7 x^{4}$

$5 x^{3}$

$20 x^{2}$

$0 x$

$10$

$7 x^{4}$

$7 x^{3}$

$21 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $- 7 x^{4} + 7 x^{3} + 21 x^{2}$

$7 x^{2}$

$x^{2}$

$x$

$3$

$7 x^{4}$

$5 x^{3}$

$20 x^{2}$

$0 x$

$10$

$7 x^{4}$

$7 x^{3}$

$21 x^{2}$

Step 5

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

$7 x^{2}$

$x^{2}$

$x$

$3$

$7 x^{4}$

$5 x^{3}$

$20 x^{2}$

$0 x$

$10$

$7 x^{4}$

$7 x^{3}$

$21 x^{2}$

$2 x^{3}$

$x^{2}$

Step 6

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

$7 x^{2}$

$x^{2}$

$x$

$3$

$7 x^{4}$

$5 x^{3}$

$20 x^{2}$

$0 x$

$10$

$7 x^{4}$

$7 x^{3}$

$21 x^{2}$

$2 x^{3}$

$x^{2}$

$0 x$

Step 7

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

$7 x^{2}$

$2 x$

$x^{2}$

$x$

$3$

$7 x^{4}$

$5 x^{3}$

$20 x^{2}$

$0 x$

$10$

$7 x^{4}$

$7 x^{3}$

$21 x^{2}$

$2 x^{3}$

$x^{2}$

$0 x$

Step 8

Multiply the new quotient term by the divisor.

$7 x^{2}$

$2 x$

$x^{2}$

$x$

$3$

$7 x^{4}$

$5 x^{3}$

$20 x^{2}$

$0 x$

$10$

$7 x^{4}$

$7 x^{3}$

$21 x^{2}$

$2 x^{3}$

$x^{2}$

$0 x$

$2 x^{3}$

$2 x^{2}$

$6 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 2 x^{3} + 2 x^{2} + 6 x$

$7 x^{2}$

$2 x$

$x^{2}$

$x$

$3$

$7 x^{4}$

$5 x^{3}$

$20 x^{2}$

$0 x$

$10$

$7 x^{4}$

$7 x^{3}$

$21 x^{2}$

$2 x^{3}$

$x^{2}$

$0 x$

$2 x^{3}$

$2 x^{2}$

$6 x$

Step 10

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

$7 x^{2}$

$2 x$

$x^{2}$

$x$

$3$

$7 x^{4}$

$5 x^{3}$

$20 x^{2}$

$0 x$

$10$

$7 x^{4}$

$7 x^{3}$

$21 x^{2}$

$2 x^{3}$

$x^{2}$

$0 x$

$2 x^{3}$

$2 x^{2}$

$6 x$

$3 x^{2}$

$6 x$

Step 11

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

$7 x^{2}$

$2 x$

$x^{2}$

$x$

$3$

$7 x^{4}$

$5 x^{3}$

$20 x^{2}$

$0 x$

$10$

$7 x^{4}$

$7 x^{3}$

$21 x^{2}$

$2 x^{3}$

$x^{2}$

$0 x$

$2 x^{3}$

$2 x^{2}$

$6 x$

$3 x^{2}$

$6 x$

$10$

Step 12

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

$7 x^{2}$

$2 x$

$3$

$x^{2}$

$x$

$3$

$7 x^{4}$

$5 x^{3}$

$20 x^{2}$

$0 x$

$10$

$7 x^{4}$

$7 x^{3}$

$21 x^{2}$

$2 x^{3}$

$x^{2}$

$0 x$

$2 x^{3}$

$2 x^{2}$

$6 x$

$3 x^{2}$

$6 x$

$10$

Step 13

Multiply the new quotient term by the divisor.

$7 x^{2}$

$2 x$

$3$

$x^{2}$

$x$

$3$

$7 x^{4}$

$5 x^{3}$

$20 x^{2}$

$0 x$

$10$

$7 x^{4}$

$7 x^{3}$

$21 x^{2}$

$2 x^{3}$

$x^{2}$

$0 x$

$2 x^{3}$

$2 x^{2}$

$6 x$

$3 x^{2}$

$6 x$

$10$

$3 x^{2}$

$3 x$

$9$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 3 x^{2} + 3 x + 9$

$7 x^{2}$

$2 x$

$3$

$x^{2}$

$x$

$3$

$7 x^{4}$

$5 x^{3}$

$20 x^{2}$

$0 x$

$10$

$7 x^{4}$

$7 x^{3}$

$21 x^{2}$

$2 x^{3}$

$x^{2}$

$0 x$

$2 x^{3}$

$2 x^{2}$

$6 x$

$3 x^{2}$

$6 x$

$10$

$3 x^{2}$

$3 x$

$9$

Step 15

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

$7 x^{2}$

$2 x$

$3$

$x^{2}$

$x$

$3$

$7 x^{4}$

$5 x^{3}$

$20 x^{2}$

$0 x$

$10$

$7 x^{4}$

$7 x^{3}$

$21 x^{2}$

$2 x^{3}$

$x^{2}$

$0 x$

$2 x^{3}$

$2 x^{2}$

$6 x$

$3 x^{2}$

$6 x$

$10$

$3 x^{2}$

$3 x$

$9$

$9 x$

$1$

Step 16

The quotient is $7 x^{2} + 2 x + 3$ .

$7 x^{2} + 2 x + 3$