Precalculus Examples

(9 x^{3} - 21 x^{2} + 18 x - 29) \div (3 x - 6)

$(9 x^{3} - 21 x^{2} + 18 x - 29) \div (3 x - 6)$

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

$0$ .

3 x

$3 x$

6

$6$

9 x^{3}

$9 x^{3}$

21 x^{2}

$21 x^{2}$

18 x

$18 x$

29

$29$

Step 2

Divide the highest order term in the dividend

9 x^{3}

$9 x^{3}$ by the highest order term in divisor

3 x

$3 x$ .

					$3 x^{2}$ $3 x^{2}$
	$3 x$ $3 x$	-	$6$ $6$		$9 x^{3}$ $9 x^{3}$	-	$21 x^{2}$ $21 x^{2}$	+	$18 x$ $18 x$	-	$29$ $29$

Step 3

Multiply the new quotient term by the divisor.

				$3 x^{2}$ $3 x^{2}$
$3 x$ $3 x$	-	$6$ $6$		$9 x^{3}$ $9 x^{3}$	-	$21 x^{2}$ $21 x^{2}$	+	$18 x$ $18 x$	-	$29$ $29$
			+	$9 x^{3}$ $9 x^{3}$	-	$18 x^{2}$ $18 x^{2}$

Step 4

The expression needs to be subtracted from the dividend, so change all the signs in

9 x^{3} - 18 x^{2}

$9 x^{3} - 18 x^{2}$

				$3 x^{2}$ $3 x^{2}$
$3 x$ $3 x$	-	$6$ $6$		$9 x^{3}$ $9 x^{3}$	-	$21 x^{2}$ $21 x^{2}$	+	$18 x$ $18 x$	-	$29$ $29$
			-	$9 x^{3}$ $9 x^{3}$	+	$18 x^{2}$ $18 x^{2}$

Step 5

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

				$3 x^{2}$ $3 x^{2}$
$3 x$ $3 x$	-	$6$ $6$		$9 x^{3}$ $9 x^{3}$	-	$21 x^{2}$ $21 x^{2}$	+	$18 x$ $18 x$	-	$29$ $29$
			-	$9 x^{3}$ $9 x^{3}$	+	$18 x^{2}$ $18 x^{2}$
					-	$3 x^{2}$ $3 x^{2}$

Step 6

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

				$3 x^{2}$ $3 x^{2}$
$3 x$ $3 x$	-	$6$ $6$		$9 x^{3}$ $9 x^{3}$	-	$21 x^{2}$ $21 x^{2}$	+	$18 x$ $18 x$	-	$29$ $29$
			-	$9 x^{3}$ $9 x^{3}$	+	$18 x^{2}$ $18 x^{2}$
					-	$3 x^{2}$ $3 x^{2}$	+	$18 x$ $18 x$

Step 7

Divide the highest order term in the dividend

- 3 x^{2}

$- 3 x^{2}$ by the highest order term in divisor

3 x

$3 x$ .

				$3 x^{2}$ $3 x^{2}$	-	$x$ $x$
$3 x$ $3 x$	-	$6$ $6$		$9 x^{3}$ $9 x^{3}$	-	$21 x^{2}$ $21 x^{2}$	+	$18 x$ $18 x$	-	$29$ $29$
			-	$9 x^{3}$ $9 x^{3}$	+	$18 x^{2}$ $18 x^{2}$
					-	$3 x^{2}$ $3 x^{2}$	+	$18 x$ $18 x$

Step 8

Multiply the new quotient term by the divisor.

				$3 x^{2}$ $3 x^{2}$	-	$x$ $x$
$3 x$ $3 x$	-	$6$ $6$		$9 x^{3}$ $9 x^{3}$	-	$21 x^{2}$ $21 x^{2}$	+	$18 x$ $18 x$	-	$29$ $29$
			-	$9 x^{3}$ $9 x^{3}$	+	$18 x^{2}$ $18 x^{2}$
					-	$3 x^{2}$ $3 x^{2}$	+	$18 x$ $18 x$
					-	$3 x^{2}$ $3 x^{2}$	+	$6 x$ $6 x$

Step 9

The expression needs to be subtracted from the dividend, so change all the signs in

- 3 x^{2} + 6 x

$- 3 x^{2} + 6 x$

				$3 x^{2}$ $3 x^{2}$	-	$x$ $x$
$3 x$ $3 x$	-	$6$ $6$		$9 x^{3}$ $9 x^{3}$	-	$21 x^{2}$ $21 x^{2}$	+	$18 x$ $18 x$	-	$29$ $29$
			-	$9 x^{3}$ $9 x^{3}$	+	$18 x^{2}$ $18 x^{2}$
					-	$3 x^{2}$ $3 x^{2}$	+	$18 x$ $18 x$
					+	$3 x^{2}$ $3 x^{2}$	-	$6 x$ $6 x$

Step 10

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

				$3 x^{2}$ $3 x^{2}$	-	$x$ $x$
$3 x$ $3 x$	-	$6$ $6$		$9 x^{3}$ $9 x^{3}$	-	$21 x^{2}$ $21 x^{2}$	+	$18 x$ $18 x$	-	$29$ $29$
			-	$9 x^{3}$ $9 x^{3}$	+	$18 x^{2}$ $18 x^{2}$
					-	$3 x^{2}$ $3 x^{2}$	+	$18 x$ $18 x$
					+	$3 x^{2}$ $3 x^{2}$	-	$6 x$ $6 x$
							+	$12 x$ $12 x$

Step 11

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

				$3 x^{2}$ $3 x^{2}$	-	$x$ $x$
$3 x$ $3 x$	-	$6$ $6$		$9 x^{3}$ $9 x^{3}$	-	$21 x^{2}$ $21 x^{2}$	+	$18 x$ $18 x$	-	$29$ $29$
			-	$9 x^{3}$ $9 x^{3}$	+	$18 x^{2}$ $18 x^{2}$
					-	$3 x^{2}$ $3 x^{2}$	+	$18 x$ $18 x$
					+	$3 x^{2}$ $3 x^{2}$	-	$6 x$ $6 x$
							+	$12 x$ $12 x$	-	$29$ $29$

Step 12

Divide the highest order term in the dividend

12 x

$12 x$ by the highest order term in divisor

3 x

$3 x$ .

				$3 x^{2}$ $3 x^{2}$	-	$x$ $x$	+	$4$ $4$
$3 x$ $3 x$	-	$6$ $6$		$9 x^{3}$ $9 x^{3}$	-	$21 x^{2}$ $21 x^{2}$	+	$18 x$ $18 x$	-	$29$ $29$
			-	$9 x^{3}$ $9 x^{3}$	+	$18 x^{2}$ $18 x^{2}$
					-	$3 x^{2}$ $3 x^{2}$	+	$18 x$ $18 x$
					+	$3 x^{2}$ $3 x^{2}$	-	$6 x$ $6 x$
							+	$12 x$ $12 x$	-	$29$ $29$

Step 13

Multiply the new quotient term by the divisor.

				$3 x^{2}$ $3 x^{2}$	-	$x$ $x$	+	$4$ $4$
$3 x$ $3 x$	-	$6$ $6$		$9 x^{3}$ $9 x^{3}$	-	$21 x^{2}$ $21 x^{2}$	+	$18 x$ $18 x$	-	$29$ $29$
			-	$9 x^{3}$ $9 x^{3}$	+	$18 x^{2}$ $18 x^{2}$
					-	$3 x^{2}$ $3 x^{2}$	+	$18 x$ $18 x$
					+	$3 x^{2}$ $3 x^{2}$	-	$6 x$ $6 x$
							+	$12 x$ $12 x$	-	$29$ $29$
							+	$12 x$ $12 x$	-	$24$ $24$

Step 14

The expression needs to be subtracted from the dividend, so change all the signs in

12 x - 24

$12 x - 24$

				$3 x^{2}$ $3 x^{2}$	-	$x$ $x$	+	$4$ $4$
$3 x$ $3 x$	-	$6$ $6$		$9 x^{3}$ $9 x^{3}$	-	$21 x^{2}$ $21 x^{2}$	+	$18 x$ $18 x$	-	$29$ $29$
			-	$9 x^{3}$ $9 x^{3}$	+	$18 x^{2}$ $18 x^{2}$
					-	$3 x^{2}$ $3 x^{2}$	+	$18 x$ $18 x$
					+	$3 x^{2}$ $3 x^{2}$	-	$6 x$ $6 x$
							+	$12 x$ $12 x$	-	$29$ $29$
							-	$12 x$ $12 x$	+	$24$ $24$

Step 15

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

				$3 x^{2}$ $3 x^{2}$	-	$x$ $x$	+	$4$ $4$
$3 x$ $3 x$	-	$6$ $6$		$9 x^{3}$ $9 x^{3}$	-	$21 x^{2}$ $21 x^{2}$	+	$18 x$ $18 x$	-	$29$ $29$
			-	$9 x^{3}$ $9 x^{3}$	+	$18 x^{2}$ $18 x^{2}$
					-	$3 x^{2}$ $3 x^{2}$	+	$18 x$ $18 x$
					+	$3 x^{2}$ $3 x^{2}$	-	$6 x$ $6 x$
							+	$12 x$ $12 x$	-	$29$ $29$
							-	$12 x$ $12 x$	+	$24$ $24$
									-	$5$ $5$

Step 16

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

3 x^{2} - x + 4 - \frac{5}{3 x - 6}

$3 x^{2} - x + 4 - \frac{5}{3 x - 6}$

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