Pre-Algebra Examples

$\frac{6 x^{3} - 5 x^{2} + 15 x - 5}{2 x^{2} - x + 3}$

Step 1

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

$2 x^{2}$

$x$

$3$

$6 x^{3}$

$5 x^{2}$

$15 x$

$5$

Step 2

Divide the highest order term in the dividend $6 x^{3}$ by the highest order term in divisor $2 x^{2}$ .

$3 x$

$2 x^{2}$

$x$

$3$

$6 x^{3}$

$5 x^{2}$

$15 x$

$5$

Step 3

Multiply the new quotient term by the divisor.

$3 x$

$2 x^{2}$

$x$

$3$

$6 x^{3}$

$5 x^{2}$

$15 x$

$5$

$6 x^{3}$

$3 x^{2}$

$9 x$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $6 x^{3} - 3 x^{2} + 9 x$

$3 x$

$2 x^{2}$

$x$

$3$

$6 x^{3}$

$5 x^{2}$

$15 x$

$5$

$6 x^{3}$

$3 x^{2}$

$9 x$

Step 5

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

$3 x$

$2 x^{2}$

$x$

$3$

$6 x^{3}$

$5 x^{2}$

$15 x$

$5$

$6 x^{3}$

$3 x^{2}$

$9 x$

$2 x^{2}$

$6 x$

Step 6

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

						$3 x$
$2 x^{2}$	-	$x$	+	$3$		$6 x^{3}$	-	$5 x^{2}$	+	$15 x$	-	$5$
					-	$6 x^{3}$	+	$3 x^{2}$	-	$9 x$
							-	$2 x^{2}$	+	$6 x$	-	$5$

Step 7

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

						$3 x$	-	$1$
$2 x^{2}$	-	$x$	+	$3$		$6 x^{3}$	-	$5 x^{2}$	+	$15 x$	-	$5$
					-	$6 x^{3}$	+	$3 x^{2}$	-	$9 x$
							-	$2 x^{2}$	+	$6 x$	-	$5$

Step 8

Multiply the new quotient term by the divisor.

						$3 x$	-	$1$
$2 x^{2}$	-	$x$	+	$3$		$6 x^{3}$	-	$5 x^{2}$	+	$15 x$	-	$5$
					-	$6 x^{3}$	+	$3 x^{2}$	-	$9 x$
							-	$2 x^{2}$	+	$6 x$	-	$5$
							-	$2 x^{2}$	+	$x$	-	$3$

Step 9

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

						$3 x$	-	$1$
$2 x^{2}$	-	$x$	+	$3$		$6 x^{3}$	-	$5 x^{2}$	+	$15 x$	-	$5$
					-	$6 x^{3}$	+	$3 x^{2}$	-	$9 x$
							-	$2 x^{2}$	+	$6 x$	-	$5$
							+	$2 x^{2}$	-	$x$	+	$3$

Step 10

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

						$3 x$	-	$1$
$2 x^{2}$	-	$x$	+	$3$		$6 x^{3}$	-	$5 x^{2}$	+	$15 x$	-	$5$
					-	$6 x^{3}$	+	$3 x^{2}$	-	$9 x$
							-	$2 x^{2}$	+	$6 x$	-	$5$
							+	$2 x^{2}$	-	$x$	+	$3$
									+	$5 x$	-	$2$

Step 11

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

$3 x - 1 + \frac{5 x - 2}{2 x^{2} - x + 3}$