Algebra Examples

$(15 x^{3} + x^{2} - 30 x - 2) \div (5 x^{2} - 10)$

Step 1

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

$5 x^{2}$

$0 x$

$10$

$15 x^{3}$

$x^{2}$

$30 x$

$2$

Step 2

Divide the highest order term in the dividend $15 x^{3}$ by the highest order term in divisor $5 x^{2}$ .

$3 x$

$5 x^{2}$

$0 x$

$10$

$15 x^{3}$

$x^{2}$

$30 x$

$2$

Step 3

Multiply the new quotient term by the divisor.

$3 x$

$5 x^{2}$

$0 x$

$10$

$15 x^{3}$

$x^{2}$

$30 x$

$2$

$15 x^{3}$

$0$

$30 x$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $15 x^{3} + 0 - 30 x$

						$3 x$
$5 x^{2}$	+	$0 x$	-	$10$		$15 x^{3}$	+	$x^{2}$	-	$30 x$	-	$2$
					-	$15 x^{3}$	-	$0$	+	$30 x$

Step 5

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

$3 x$

$5 x^{2}$

$0 x$

$10$

$15 x^{3}$

$x^{2}$

$30 x$

$2$

$15 x^{3}$

$0$

$30 x$

$x^{2}$

$0$

Step 6

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

						$3 x$
$5 x^{2}$	+	$0 x$	-	$10$		$15 x^{3}$	+	$x^{2}$	-	$30 x$	-	$2$
					-	$15 x^{3}$	-	$0$	+	$30 x$
							+	$x^{2}$	+	$0$	-	$2$

Step 7

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

						$3 x$	+	$\frac{1}{5}$
$5 x^{2}$	+	$0 x$	-	$10$		$15 x^{3}$	+	$x^{2}$	-	$30 x$	-	$2$
					-	$15 x^{3}$	-	$0$	+	$30 x$
							+	$x^{2}$	+	$0$	-	$2$

Step 8

Multiply the new quotient term by the divisor.

						$3 x$	+	$\frac{1}{5}$
$5 x^{2}$	+	$0 x$	-	$10$		$15 x^{3}$	+	$x^{2}$	-	$30 x$	-	$2$
					-	$15 x^{3}$	-	$0$	+	$30 x$
							+	$x^{2}$	+	$0$	-	$2$
							+	$x^{2}$	+	$0$	-	$2$

Step 9

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

						$3 x$	+	$\frac{1}{5}$
$5 x^{2}$	+	$0 x$	-	$10$		$15 x^{3}$	+	$x^{2}$	-	$30 x$	-	$2$
					-	$15 x^{3}$	-	$0$	+	$30 x$
							+	$x^{2}$	+	$0$	-	$2$
							-	$x^{2}$	-	$0$	+	$2$

Step 10

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

						$3 x$	+	$\frac{1}{5}$
$5 x^{2}$	+	$0 x$	-	$10$		$15 x^{3}$	+	$x^{2}$	-	$30 x$	-	$2$
					-	$15 x^{3}$	-	$0$	+	$30 x$
							+	$x^{2}$	+	$0$	-	$2$
							-	$x^{2}$	-	$0$	+	$2$
												$0$

Step 11

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

$3 x + \frac{1}{5}$