Algebra Examples

$(2 x^{5} + x^{4} - 15 x^{3} - 2 x^{2} + 10 x - 24) \div (x^{2} - x - 4)$

Step 1

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

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

Step 2

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

							$2 x^{3}$
	$x^{2}$	-	$x$	-	$4$		$2 x^{5}$	+	$x^{4}$	-	$15 x^{3}$	-	$2 x^{2}$	+	$10 x$	-	$24$

Step 3

Multiply the new quotient term by the divisor.

$2 x^{3}$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{5} - 2 x^{4} - 8 x^{3}$

$2 x^{3}$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

Step 5

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

$2 x^{3}$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

$3 x^{4}$

$7 x^{3}$

Step 6

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

$2 x^{3}$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

Step 7

Divide the highest order term in the dividend $3 x^{4}$ by the highest order term in divisor $x^{2}$ .

$2 x^{3}$

$3 x^{2}$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

Step 8

Multiply the new quotient term by the divisor.

$2 x^{3}$

$3 x^{2}$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$3 x^{4}$

$3 x^{3}$

$12 x^{2}$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $3 x^{4} - 3 x^{3} - 12 x^{2}$

$2 x^{3}$

$3 x^{2}$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$3 x^{4}$

$3 x^{3}$

$12 x^{2}$

Step 10

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

$2 x^{3}$

$3 x^{2}$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$3 x^{4}$

$3 x^{3}$

$12 x^{2}$

$4 x^{3}$

$10 x^{2}$

Step 11

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

$2 x^{3}$

$3 x^{2}$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$3 x^{4}$

$3 x^{3}$

$12 x^{2}$

$4 x^{3}$

$10 x^{2}$

$10 x$

Step 12

Divide the highest order term in the dividend $- 4 x^{3}$ by the highest order term in divisor $x^{2}$ .

$2 x^{3}$

$3 x^{2}$

$4 x$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$3 x^{4}$

$3 x^{3}$

$12 x^{2}$

$4 x^{3}$

$10 x^{2}$

$10 x$

Step 13

Multiply the new quotient term by the divisor.

$2 x^{3}$

$3 x^{2}$

$4 x$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$3 x^{4}$

$3 x^{3}$

$12 x^{2}$

$4 x^{3}$

$10 x^{2}$

$10 x$

$4 x^{3}$

$4 x^{2}$

$16 x$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{3} + 4 x^{2} + 16 x$

$2 x^{3}$

$3 x^{2}$

$4 x$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$3 x^{4}$

$3 x^{3}$

$12 x^{2}$

$4 x^{3}$

$10 x^{2}$

$10 x$

$4 x^{3}$

$4 x^{2}$

$16 x$

Step 15

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

$2 x^{3}$

$3 x^{2}$

$4 x$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$3 x^{4}$

$3 x^{3}$

$12 x^{2}$

$4 x^{3}$

$10 x^{2}$

$10 x$

$4 x^{3}$

$4 x^{2}$

$16 x$

$6 x^{2}$

$6 x$

Step 16

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

$2 x^{3}$

$3 x^{2}$

$4 x$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$3 x^{4}$

$3 x^{3}$

$12 x^{2}$

$4 x^{3}$

$10 x^{2}$

$10 x$

$4 x^{3}$

$4 x^{2}$

$16 x$

$6 x^{2}$

$6 x$

$24$

Step 17

Divide the highest order term in the dividend $6 x^{2}$ by the highest order term in divisor $x^{2}$ .

$2 x^{3}$

$3 x^{2}$

$4 x$

$6$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$3 x^{4}$

$3 x^{3}$

$12 x^{2}$

$4 x^{3}$

$10 x^{2}$

$10 x$

$4 x^{3}$

$4 x^{2}$

$16 x$

$6 x^{2}$

$6 x$

$24$

Step 18

Multiply the new quotient term by the divisor.

$2 x^{3}$

$3 x^{2}$

$4 x$

$6$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$3 x^{4}$

$3 x^{3}$

$12 x^{2}$

$4 x^{3}$

$10 x^{2}$

$10 x$

$4 x^{3}$

$4 x^{2}$

$16 x$

$6 x^{2}$

$6 x$

$24$

$6 x^{2}$

$6 x$

$24$

Step 19

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{2} - 6 x - 24$

$2 x^{3}$

$3 x^{2}$

$4 x$

$6$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$3 x^{4}$

$3 x^{3}$

$12 x^{2}$

$4 x^{3}$

$10 x^{2}$

$10 x$

$4 x^{3}$

$4 x^{2}$

$16 x$

$6 x^{2}$

$6 x$

$24$

$6 x^{2}$

$6 x$

$24$

Step 20

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

$2 x^{3}$

$3 x^{2}$

$4 x$

$6$

$x^{2}$

$x$

$4$

$2 x^{5}$

$x^{4}$

$15 x^{3}$

$2 x^{2}$

$10 x$

$24$

$2 x^{5}$

$2 x^{4}$

$8 x^{3}$

$3 x^{4}$

$7 x^{3}$

$2 x^{2}$

$3 x^{4}$

$3 x^{3}$

$12 x^{2}$

$4 x^{3}$

$10 x^{2}$

$10 x$

$4 x^{3}$

$4 x^{2}$

$16 x$

$6 x^{2}$

$6 x$

$24$

$6 x^{2}$

$6 x$

$24$

$0$

Step 21

Since the remander is $0$ , the final answer is the quotient.

$2 x^{3} + 3 x^{2} - 4 x + 6$