Pre-Algebra Examples

$(5 x^{4} - 2 x^{3} - 7 x^{2} - 39) \div (x^{2} + 2 x - 4)$

Step 1

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

$x^{2}$

$2 x$

$4$

$5 x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$39$

Step 2

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

							$5 x^{2}$
	$x^{2}$	+	$2 x$	-	$4$		$5 x^{4}$	-	$2 x^{3}$	-	$7 x^{2}$	+	$0 x$	-	$39$

Step 3

Multiply the new quotient term by the divisor.

$5 x^{2}$

$x^{2}$

$2 x$

$4$

$5 x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$39$

$5 x^{4}$

$10 x^{3}$

$20 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{4} + 10 x^{3} - 20 x^{2}$

$5 x^{2}$

$x^{2}$

$2 x$

$4$

$5 x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$39$

$5 x^{4}$

$10 x^{3}$

$20 x^{2}$

Step 5

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

$5 x^{2}$

$x^{2}$

$2 x$

$4$

$5 x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$39$

$5 x^{4}$

$10 x^{3}$

$20 x^{2}$

$12 x^{3}$

$13 x^{2}$

Step 6

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

$5 x^{2}$

$x^{2}$

$2 x$

$4$

$5 x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$39$

$5 x^{4}$

$10 x^{3}$

$20 x^{2}$

$12 x^{3}$

$13 x^{2}$

$0 x$

Step 7

Divide the highest order term in the dividend $- 12 x^{3}$ by the highest order term in divisor $x^{2}$ .

$5 x^{2}$

$12 x$

$x^{2}$

$2 x$

$4$

$5 x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$39$

$5 x^{4}$

$10 x^{3}$

$20 x^{2}$

$12 x^{3}$

$13 x^{2}$

$0 x$

Step 8

Multiply the new quotient term by the divisor.

$5 x^{2}$

$12 x$

$x^{2}$

$2 x$

$4$

$5 x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$39$

$5 x^{4}$

$10 x^{3}$

$20 x^{2}$

$12 x^{3}$

$13 x^{2}$

$0 x$

$12 x^{3}$

$24 x^{2}$

$48 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in $- 12 x^{3} - 24 x^{2} + 48 x$

$5 x^{2}$

$12 x$

$x^{2}$

$2 x$

$4$

$5 x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$39$

$5 x^{4}$

$10 x^{3}$

$20 x^{2}$

$12 x^{3}$

$13 x^{2}$

$0 x$

$12 x^{3}$

$24 x^{2}$

$48 x$

Step 10

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

$5 x^{2}$

$12 x$

$x^{2}$

$2 x$

$4$

$5 x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$39$

$5 x^{4}$

$10 x^{3}$

$20 x^{2}$

$12 x^{3}$

$13 x^{2}$

$0 x$

$12 x^{3}$

$24 x^{2}$

$48 x$

$37 x^{2}$

$48 x$

Step 11

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

$5 x^{2}$

$12 x$

$x^{2}$

$2 x$

$4$

$5 x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$39$

$5 x^{4}$

$10 x^{3}$

$20 x^{2}$

$12 x^{3}$

$13 x^{2}$

$0 x$

$12 x^{3}$

$24 x^{2}$

$48 x$

$37 x^{2}$

$48 x$

$39$

Step 12

Divide the highest order term in the dividend $37 x^{2}$ by the highest order term in divisor $x^{2}$ .

$5 x^{2}$

$12 x$

$37$

$x^{2}$

$2 x$

$4$

$5 x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$39$

$5 x^{4}$

$10 x^{3}$

$20 x^{2}$

$12 x^{3}$

$13 x^{2}$

$0 x$

$12 x^{3}$

$24 x^{2}$

$48 x$

$37 x^{2}$

$48 x$

$39$

Step 13

Multiply the new quotient term by the divisor.

$5 x^{2}$

$12 x$

$37$

$x^{2}$

$2 x$

$4$

$5 x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$39$

$5 x^{4}$

$10 x^{3}$

$20 x^{2}$

$12 x^{3}$

$13 x^{2}$

$0 x$

$12 x^{3}$

$24 x^{2}$

$48 x$

$37 x^{2}$

$48 x$

$39$

$37 x^{2}$

$74 x$

$148$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in $37 x^{2} + 74 x - 148$

$5 x^{2}$

$12 x$

$37$

$x^{2}$

$2 x$

$4$

$5 x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$39$

$5 x^{4}$

$10 x^{3}$

$20 x^{2}$

$12 x^{3}$

$13 x^{2}$

$0 x$

$12 x^{3}$

$24 x^{2}$

$48 x$

$37 x^{2}$

$48 x$

$39$

$37 x^{2}$

$74 x$

$148$

Step 15

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

$5 x^{2}$

$12 x$

$37$

$x^{2}$

$2 x$

$4$

$5 x^{4}$

$2 x^{3}$

$7 x^{2}$

$0 x$

$39$

$5 x^{4}$

$10 x^{3}$

$20 x^{2}$

$12 x^{3}$

$13 x^{2}$

$0 x$

$12 x^{3}$

$24 x^{2}$

$48 x$

$37 x^{2}$

$48 x$

$39$

$37 x^{2}$

$74 x$

$148$

$122 x$

$109$

Step 16

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

$5 x^{2} - 12 x + 37 + \frac{- 122 x + 109}{x^{2} + 2 x - 4}$