Algebra Examples

$(2 x^{4} - 5 x^{3} - 15 x^{2} + 8 x + 16) \div (x^{2} - 3 x - 4)$

Step 1

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

$x^{2}$

$3 x$

$4$

$2 x^{4}$

$5 x^{3}$

$15 x^{2}$

$8 x$

$16$

Step 2

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

							$2 x^{2}$
	$x^{2}$	-	$3 x$	-	$4$		$2 x^{4}$	-	$5 x^{3}$	-	$15 x^{2}$	+	$8 x$	+	$16$

Step 3

Multiply the new quotient term by the divisor.

$2 x^{2}$

$x^{2}$

$3 x$

$4$

$2 x^{4}$

$5 x^{3}$

$15 x^{2}$

$8 x$

$16$

$2 x^{4}$

$6 x^{3}$

$8 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{4} - 6 x^{3} - 8 x^{2}$

$2 x^{2}$

$x^{2}$

$3 x$

$4$

$2 x^{4}$

$5 x^{3}$

$15 x^{2}$

$8 x$

$16$

$2 x^{4}$

$6 x^{3}$

$8 x^{2}$

Step 5

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

$2 x^{2}$

$x^{2}$

$3 x$

$4$

$2 x^{4}$

$5 x^{3}$

$15 x^{2}$

$8 x$

$16$

$2 x^{4}$

$6 x^{3}$

$8 x^{2}$

$x^{3}$

$7 x^{2}$

Step 6

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

$2 x^{2}$

$x^{2}$

$3 x$

$4$

$2 x^{4}$

$5 x^{3}$

$15 x^{2}$

$8 x$

$16$

$2 x^{4}$

$6 x^{3}$

$8 x^{2}$

$x^{3}$

$7 x^{2}$

$8 x$

Step 7

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

$2 x^{2}$

$x$

$x^{2}$

$3 x$

$4$

$2 x^{4}$

$5 x^{3}$

$15 x^{2}$

$8 x$

$16$

$2 x^{4}$

$6 x^{3}$

$8 x^{2}$

$x^{3}$

$7 x^{2}$

$8 x$

Step 8

Multiply the new quotient term by the divisor.

$2 x^{2}$

$x$

$x^{2}$

$3 x$

$4$

$2 x^{4}$

$5 x^{3}$

$15 x^{2}$

$8 x$

$16$

$2 x^{4}$

$6 x^{3}$

$8 x^{2}$

$x^{3}$

$7 x^{2}$

$8 x$

$x^{3}$

$3 x^{2}$

$4 x$

Step 9

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

$2 x^{2}$

$x$

$x^{2}$

$3 x$

$4$

$2 x^{4}$

$5 x^{3}$

$15 x^{2}$

$8 x$

$16$

$2 x^{4}$

$6 x^{3}$

$8 x^{2}$

$x^{3}$

$7 x^{2}$

$8 x$

$x^{3}$

$3 x^{2}$

$4 x$

Step 10

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

$2 x^{2}$

$x$

$x^{2}$

$3 x$

$4$

$2 x^{4}$

$5 x^{3}$

$15 x^{2}$

$8 x$

$16$

$2 x^{4}$

$6 x^{3}$

$8 x^{2}$

$x^{3}$

$7 x^{2}$

$8 x$

$x^{3}$

$3 x^{2}$

$4 x$

$4 x^{2}$

$12 x$

Step 11

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

$2 x^{2}$

$x$

$x^{2}$

$3 x$

$4$

$2 x^{4}$

$5 x^{3}$

$15 x^{2}$

$8 x$

$16$

$2 x^{4}$

$6 x^{3}$

$8 x^{2}$

$x^{3}$

$7 x^{2}$

$8 x$

$x^{3}$

$3 x^{2}$

$4 x$

$4 x^{2}$

$12 x$

$16$

Step 12

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

$2 x^{2}$

$x$

$4$

$x^{2}$

$3 x$

$4$

$2 x^{4}$

$5 x^{3}$

$15 x^{2}$

$8 x$

$16$

$2 x^{4}$

$6 x^{3}$

$8 x^{2}$

$x^{3}$

$7 x^{2}$

$8 x$

$x^{3}$

$3 x^{2}$

$4 x$

$4 x^{2}$

$12 x$

$16$

Step 13

Multiply the new quotient term by the divisor.

$2 x^{2}$

$x$

$4$

$x^{2}$

$3 x$

$4$

$2 x^{4}$

$5 x^{3}$

$15 x^{2}$

$8 x$

$16$

$2 x^{4}$

$6 x^{3}$

$8 x^{2}$

$x^{3}$

$7 x^{2}$

$8 x$

$x^{3}$

$3 x^{2}$

$4 x$

$4 x^{2}$

$12 x$

$16$

$4 x^{2}$

$12 x$

$16$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} + 12 x + 16$

$2 x^{2}$

$x$

$4$

$x^{2}$

$3 x$

$4$

$2 x^{4}$

$5 x^{3}$

$15 x^{2}$

$8 x$

$16$

$2 x^{4}$

$6 x^{3}$

$8 x^{2}$

$x^{3}$

$7 x^{2}$

$8 x$

$x^{3}$

$3 x^{2}$

$4 x$

$4 x^{2}$

$12 x$

$16$

$4 x^{2}$

$12 x$

$16$

Step 15

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

$2 x^{2}$

$x$

$4$

$x^{2}$

$3 x$

$4$

$2 x^{4}$

$5 x^{3}$

$15 x^{2}$

$8 x$

$16$

$2 x^{4}$

$6 x^{3}$

$8 x^{2}$

$x^{3}$

$7 x^{2}$

$8 x$

$x^{3}$

$3 x^{2}$

$4 x$

$4 x^{2}$

$12 x$

$16$

$4 x^{2}$

$12 x$

$16$

$0$

Step 16

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

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