Precalculus Examples

\frac{3 x + 6 x^{2}}{x + 3}

$\frac{3 x + 6 x^{2}}{x + 3}$

Step 1

To calculate the remainder, first divide the polynomials.

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Step 1.1

Reorder

3 x

$3 x$ and

6 x^{2}

$6 x^{2}$ .

\frac{6 x^{2} + 3 x}{x + 3}

$\frac{6 x^{2} + 3 x}{x + 3}$

Step 1.2

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

0

$0$ .

x

$x$

3

$3$

6 x^{2}

$6 x^{2}$

3 x

$3 x$

0

$0$

Step 1.3

Divide the highest order term in the dividend

6 x^{2}

$6 x^{2}$ by the highest order term in divisor

x

$x$ .

					$6 x$ $6 x$
	$x$ $x$	+	$3$ $3$		$6 x^{2}$ $6 x^{2}$	+	$3 x$ $3 x$	+	$0$ $0$

Step 1.4

Multiply the new quotient term by the divisor.

				$6 x$ $6 x$
$x$ $x$	+	$3$ $3$		$6 x^{2}$ $6 x^{2}$	+	$3 x$ $3 x$	+	$0$ $0$
			+	$6 x^{2}$ $6 x^{2}$	+	$18 x$ $18 x$

Step 1.5

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

6 x^{2} + 18 x

$6 x^{2} + 18 x$

				$6 x$ $6 x$
$x$ $x$	+	$3$ $3$		$6 x^{2}$ $6 x^{2}$	+	$3 x$ $3 x$	+	$0$ $0$
			-	$6 x^{2}$ $6 x^{2}$	-	$18 x$ $18 x$

Step 1.6

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

				$6 x$ $6 x$
$x$ $x$	+	$3$ $3$		$6 x^{2}$ $6 x^{2}$	+	$3 x$ $3 x$	+	$0$ $0$
			-	$6 x^{2}$ $6 x^{2}$	-	$18 x$ $18 x$
					-	$15 x$ $15 x$

Step 1.7

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

				$6 x$ $6 x$
$x$ $x$	+	$3$ $3$		$6 x^{2}$ $6 x^{2}$	+	$3 x$ $3 x$	+	$0$ $0$
			-	$6 x^{2}$ $6 x^{2}$	-	$18 x$ $18 x$
					-	$15 x$ $15 x$	+	$0$ $0$

Step 1.8

Divide the highest order term in the dividend

- 15 x

$- 15 x$ by the highest order term in divisor

x

$x$ .

				$6 x$ $6 x$	-	$15$ $15$
$x$ $x$	+	$3$ $3$		$6 x^{2}$ $6 x^{2}$	+	$3 x$ $3 x$	+	$0$ $0$
			-	$6 x^{2}$ $6 x^{2}$	-	$18 x$ $18 x$
					-	$15 x$ $15 x$	+	$0$ $0$

Step 1.9

Multiply the new quotient term by the divisor.

				$6 x$ $6 x$	-	$15$ $15$
$x$ $x$	+	$3$ $3$		$6 x^{2}$ $6 x^{2}$	+	$3 x$ $3 x$	+	$0$ $0$
			-	$6 x^{2}$ $6 x^{2}$	-	$18 x$ $18 x$
					-	$15 x$ $15 x$	+	$0$ $0$
					-	$15 x$ $15 x$	-	$45$ $45$

Step 1.10

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

- 15 x - 45

$- 15 x - 45$

				$6 x$ $6 x$	-	$15$ $15$
$x$ $x$	+	$3$ $3$		$6 x^{2}$ $6 x^{2}$	+	$3 x$ $3 x$	+	$0$ $0$
			-	$6 x^{2}$ $6 x^{2}$	-	$18 x$ $18 x$
					-	$15 x$ $15 x$	+	$0$ $0$
					+	$15 x$ $15 x$	+	$45$ $45$

Step 1.11

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

				$6 x$ $6 x$	-	$15$ $15$
$x$ $x$	+	$3$ $3$		$6 x^{2}$ $6 x^{2}$	+	$3 x$ $3 x$	+	$0$ $0$
			-	$6 x^{2}$ $6 x^{2}$	-	$18 x$ $18 x$
					-	$15 x$ $15 x$	+	$0$ $0$
					+	$15 x$ $15 x$	+	$45$ $45$
							+	$45$ $45$

Step 1.12

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

6 x - 15 + \frac{45}{x + 3}

$6 x - 15 + \frac{45}{x + 3}$

6 x - 15 + \frac{45}{x + 3}

$6 x - 15 + \frac{45}{x + 3}$

Step 2

Since the last term in the resulting expression is a fraction, the numerator of the fraction is the remainder.

45

$45$

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