Algebra Examples

$(- 16 x + 15 x^{3} - 4 + 18 x^{2}) \div (3 x^{2} - 2)$

Step 1

Expand $- 16 x + 15 x^{3} - 4 + 18 x^{2}$ .

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Step 1.1

Reorder $- 16 x$ and $15 x^{3}$ .

$\frac{15 x^{3} - 16 x - 4 + 18 x^{2}}{3 x^{2} - 2}$

Step 1.2

Move $- 4$ .

$\frac{15 x^{3} - 16 x + 18 x^{2} - 4}{3 x^{2} - 2}$

Step 1.3

Move $- 16 x$ .

$\frac{15 x^{3} + 18 x^{2} - 16 x - 4}{3 x^{2} - 2}$

Step 2

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

$3 x^{2}$

$0 x$

$2$

$15 x^{3}$

$18 x^{2}$

$16 x$

$4$

Step 3

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

$5 x$

$3 x^{2}$

$0 x$

$2$

$15 x^{3}$

$18 x^{2}$

$16 x$

$4$

Step 4

Multiply the new quotient term by the divisor.

$5 x$

$3 x^{2}$

$0 x$

$2$

$15 x^{3}$

$18 x^{2}$

$16 x$

$4$

$15 x^{3}$

$0$

$10 x$

Step 5

The expression needs to be subtracted from the dividend, so change all the signs in $15 x^{3} + 0 - 10 x$

						$5 x$
$3 x^{2}$	+	$0 x$	-	$2$		$15 x^{3}$	+	$18 x^{2}$	-	$16 x$	-	$4$
					-	$15 x^{3}$	-	$0$	+	$10 x$

Step 6

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

$5 x$

$3 x^{2}$

$0 x$

$2$

$15 x^{3}$

$18 x^{2}$

$16 x$

$4$

$15 x^{3}$

$0$

$10 x$

$18 x^{2}$

$6 x$

Step 7

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

						$5 x$
$3 x^{2}$	+	$0 x$	-	$2$		$15 x^{3}$	+	$18 x^{2}$	-	$16 x$	-	$4$
					-	$15 x^{3}$	-	$0$	+	$10 x$
							+	$18 x^{2}$	-	$6 x$	-	$4$

Step 8

Divide the highest order term in the dividend $18 x^{2}$ by the highest order term in divisor $3 x^{2}$ .

						$5 x$	+	$6$
$3 x^{2}$	+	$0 x$	-	$2$		$15 x^{3}$	+	$18 x^{2}$	-	$16 x$	-	$4$
					-	$15 x^{3}$	-	$0$	+	$10 x$
							+	$18 x^{2}$	-	$6 x$	-	$4$

Step 9

Multiply the new quotient term by the divisor.

						$5 x$	+	$6$
$3 x^{2}$	+	$0 x$	-	$2$		$15 x^{3}$	+	$18 x^{2}$	-	$16 x$	-	$4$
					-	$15 x^{3}$	-	$0$	+	$10 x$
							+	$18 x^{2}$	-	$6 x$	-	$4$
							+	$18 x^{2}$	+	$0$	-	$12$

Step 10

The expression needs to be subtracted from the dividend, so change all the signs in $18 x^{2} + 0 - 12$

						$5 x$	+	$6$
$3 x^{2}$	+	$0 x$	-	$2$		$15 x^{3}$	+	$18 x^{2}$	-	$16 x$	-	$4$
					-	$15 x^{3}$	-	$0$	+	$10 x$
							+	$18 x^{2}$	-	$6 x$	-	$4$
							-	$18 x^{2}$	-	$0$	+	$12$

Step 11

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

						$5 x$	+	$6$
$3 x^{2}$	+	$0 x$	-	$2$		$15 x^{3}$	+	$18 x^{2}$	-	$16 x$	-	$4$
					-	$15 x^{3}$	-	$0$	+	$10 x$
							+	$18 x^{2}$	-	$6 x$	-	$4$
							-	$18 x^{2}$	-	$0$	+	$12$
									-	$6 x$	+	$8$

Step 12

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

$5 x + 6 + \frac{- 6 x + 8}{3 x^{2} - 2}$