Pre-Algebra Examples

$(5 x^{3} - 13 x^{2} - x + 2) \div (x^{2} - 3 x + 1)$

Step 1

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

$x^{2}$

$3 x$

$1$

$5 x^{3}$

$13 x^{2}$

$x$

$2$

Step 2

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

$5 x$

$x^{2}$

$3 x$

$1$

$5 x^{3}$

$13 x^{2}$

$x$

$2$

Step 3

Multiply the new quotient term by the divisor.

$5 x$

$x^{2}$

$3 x$

$1$

$5 x^{3}$

$13 x^{2}$

$x$

$2$

$5 x^{3}$

$15 x^{2}$

$5 x$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} - 15 x^{2} + 5 x$

$5 x$

$x^{2}$

$3 x$

$1$

$5 x^{3}$

$13 x^{2}$

$x$

$2$

$5 x^{3}$

$15 x^{2}$

$5 x$

Step 5

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

$5 x$

$x^{2}$

$3 x$

$1$

$5 x^{3}$

$13 x^{2}$

$x$

$2$

$5 x^{3}$

$15 x^{2}$

$5 x$

$2 x^{2}$

$6 x$

Step 6

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

$5 x$

$x^{2}$

$3 x$

$1$

$5 x^{3}$

$13 x^{2}$

$x$

$2$

$5 x^{3}$

$15 x^{2}$

$5 x$

$2 x^{2}$

$6 x$

$2$

Step 7

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

$5 x$

$2$

$x^{2}$

$3 x$

$1$

$5 x^{3}$

$13 x^{2}$

$x$

$2$

$5 x^{3}$

$15 x^{2}$

$5 x$

$2 x^{2}$

$6 x$

$2$

Step 8

Multiply the new quotient term by the divisor.

$5 x$

$2$

$x^{2}$

$3 x$

$1$

$5 x^{3}$

$13 x^{2}$

$x$

$2$

$5 x^{3}$

$15 x^{2}$

$5 x$

$2 x^{2}$

$6 x$

$2$

$2 x^{2}$

$6 x$

$2$

Step 9

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

$5 x$

$2$

$x^{2}$

$3 x$

$1$

$5 x^{3}$

$13 x^{2}$

$x$

$2$

$5 x^{3}$

$15 x^{2}$

$5 x$

$2 x^{2}$

$6 x$

$2$

$2 x^{2}$

$6 x$

$2$

Step 10

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

						$5 x$	+	$2$
$x^{2}$	-	$3 x$	+	$1$		$5 x^{3}$	-	$13 x^{2}$	-	$x$	+	$2$
					-	$5 x^{3}$	+	$15 x^{2}$	-	$5 x$
							+	$2 x^{2}$	-	$6 x$	+	$2$
							-	$2 x^{2}$	+	$6 x$	-	$2$
												$0$

Step 11

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

$5 x + 2$