Precalculus Examples

$\frac{x^{3} + 6 x^{2} - 1 x - 30}{x - 2}$

Step 1

Simplify the numerator.

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Step 1.1

Rewrite $- 1 x$ as $- x$ .

$\frac{x^{3} + 6 x^{2} - x - 30}{x - 2}$

Step 1.2

Rewrite $x^{3} + 6 x^{2} - x - 30$ in a factored form.

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Step 1.2.1

Factor $x^{3} + 6 x^{2} - x - 30$ using the rational roots test.

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Step 1.2.1.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6$

$q = \pm 1$

Step 1.2.1.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 30, \pm 2, \pm 15, \pm 3, \pm 10, \pm 5, \pm 6$

Step 1.2.1.3

Substitute $2$ and simplify the expression. In this case, the expression is equal to $0$ so $2$ is a root of the polynomial.

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Step 1.2.1.3.1

Substitute $2$ into the polynomial.

$2^{3} + 6 \cdot 2^{2} - 2 - 30$

Step 1.2.1.3.2

Raise $2$ to the power of $3$ .

$8 + 6 \cdot 2^{2} - 2 - 30$

Step 1.2.1.3.3

Raise $2$ to the power of $2$ .

$8 + 6 \cdot 4 - 2 - 30$

Step 1.2.1.3.4

Multiply $6$ by $4$ .

$8 + 24 - 2 - 30$

Step 1.2.1.3.5

Add $8$ and $24$ .

$32 - 2 - 30$

Step 1.2.1.3.6

Subtract $2$ from $32$ .

$30 - 30$

Step 1.2.1.3.7

Subtract $30$ from $30$ .

$0$

Step 1.2.1.4

Since $2$ is a known root, divide the polynomial by $x - 2$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} + 6 x^{2} - x - 30}{x - 2}$

Step 1.2.1.5

Divide $x^{3} + 6 x^{2} - x - 30$ by $x - 2$ .

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Step 1.2.1.5.1

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

$x$

$2$

$x^{3}$

$6 x^{2}$

$x$

$30$

Step 1.2.1.5.2

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

					$x^{2}$
	$x$	-	$2$		$x^{3}$	+	$6 x^{2}$	-	$x$	-	$30$

Step 1.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$2$		$x^{3}$	+	$6 x^{2}$	-	$x$	-	$30$
			+	$x^{3}$	-	$2 x^{2}$

Step 1.2.1.5.4

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	+	$6 x^{2}$	-	$x$	-	$30$
			-	$x^{3}$	+	$2 x^{2}$

Step 1.2.1.5.5

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	+	$6 x^{2}$	-	$x$	-	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$8 x^{2}$

Step 1.2.1.5.6

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

				$x^{2}$
$x$	-	$2$		$x^{3}$	+	$6 x^{2}$	-	$x$	-	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$8 x^{2}$	-	$x$

Step 1.2.1.5.7

Divide the highest order term in the dividend $8 x^{2}$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$8 x$
$x$	-	$2$		$x^{3}$	+	$6 x^{2}$	-	$x$	-	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$8 x^{2}$	-	$x$

Step 1.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$8 x$
$x$	-	$2$		$x^{3}$	+	$6 x^{2}$	-	$x$	-	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$8 x^{2}$	-	$x$
					+	$8 x^{2}$	-	$16 x$

Step 1.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $8 x^{2} - 16 x$

				$x^{2}$	+	$8 x$
$x$	-	$2$		$x^{3}$	+	$6 x^{2}$	-	$x$	-	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$8 x^{2}$	-	$x$
					-	$8 x^{2}$	+	$16 x$

Step 1.2.1.5.10

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

				$x^{2}$	+	$8 x$
$x$	-	$2$		$x^{3}$	+	$6 x^{2}$	-	$x$	-	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$8 x^{2}$	-	$x$
					-	$8 x^{2}$	+	$16 x$
							+	$15 x$

Step 1.2.1.5.11

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

				$x^{2}$	+	$8 x$
$x$	-	$2$		$x^{3}$	+	$6 x^{2}$	-	$x$	-	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$8 x^{2}$	-	$x$
					-	$8 x^{2}$	+	$16 x$
							+	$15 x$	-	$30$

Step 1.2.1.5.12

Divide the highest order term in the dividend $15 x$ by the highest order term in divisor $x$ .

				$x^{2}$	+	$8 x$	+	$15$
$x$	-	$2$		$x^{3}$	+	$6 x^{2}$	-	$x$	-	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$8 x^{2}$	-	$x$
					-	$8 x^{2}$	+	$16 x$
							+	$15 x$	-	$30$

Step 1.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$8 x$	+	$15$
$x$	-	$2$		$x^{3}$	+	$6 x^{2}$	-	$x$	-	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$8 x^{2}$	-	$x$
					-	$8 x^{2}$	+	$16 x$
							+	$15 x$	-	$30$
							+	$15 x$	-	$30$

Step 1.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $15 x - 30$

				$x^{2}$	+	$8 x$	+	$15$
$x$	-	$2$		$x^{3}$	+	$6 x^{2}$	-	$x$	-	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$8 x^{2}$	-	$x$
					-	$8 x^{2}$	+	$16 x$
							+	$15 x$	-	$30$
							-	$15 x$	+	$30$

Step 1.2.1.5.15

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

				$x^{2}$	+	$8 x$	+	$15$
$x$	-	$2$		$x^{3}$	+	$6 x^{2}$	-	$x$	-	$30$
			-	$x^{3}$	+	$2 x^{2}$
					+	$8 x^{2}$	-	$x$
					-	$8 x^{2}$	+	$16 x$
							+	$15 x$	-	$30$
							-	$15 x$	+	$30$
										$0$

Step 1.2.1.5.16

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

$x^{2} + 8 x + 15$

Step 1.2.1.6

Write $x^{3} + 6 x^{2} - x - 30$ as a set of factors.

$\frac{(x - 2) (x^{2} + 8 x + 15)}{x - 2}$

Step 1.2.2

Factor $x^{2} + 8 x + 15$ using the AC method.

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Step 1.2.2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $15$ and whose sum is $8$ .

$3, 5$

Step 1.2.2.2

Write the factored form using these integers.

$\frac{(x - 2) ((x + 3) (x + 5))}{x - 2}$

$\frac{(x - 2) (x + 3) (x + 5)}{x - 2}$

Step 2

Cancel the common factor of $x - 2$ .

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Step 2.1

Cancel the common factor.

$\frac{(x - 2) (x + 3) (x + 5)}{x - 2}$

Step 2.2

Divide $(x + 3) (x + 5)$ by $1$ .

$(x + 3) (x + 5)$

Step 3

Expand $(x + 3) (x + 5)$ using the FOIL Method.

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Step 3.1

Apply the distributive property.

$x (x + 5) + 3 (x + 5)$

Step 3.2

Apply the distributive property.

$x \cdot x + x \cdot 5 + 3 (x + 5)$

Step 3.3

Apply the distributive property.

$x \cdot x + x \cdot 5 + 3 x + 3 \cdot 5$

Step 4

Simplify and combine like terms.

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Step 4.1

Simplify each term.

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Step 4.1.1

Multiply $x$ by $x$ .

$x^{2} + x \cdot 5 + 3 x + 3 \cdot 5$

Step 4.1.2

Move $5$ to the left of $x$ .

$x^{2} + 5 \cdot x + 3 x + 3 \cdot 5$

Step 4.1.3

Multiply $3$ by $5$ .

$x^{2} + 5 x + 3 x + 15$

Step 4.2

Add $5 x$ and $3 x$ .

$x^{2} + 8 x + 15$