Precalculus Examples

$2 x^{3} - 3 x^{2} - 17 x + 30$

Step 1

Factor $2 x^{3} - 3 x^{2} - 17 x + 30$ using the rational roots test.

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Step 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.

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

$q = \pm 1, \pm 2$

Step 1.2

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

$\pm 1, \pm 0.5, \pm 30, \pm 15, \pm 2, \pm 7.5, \pm 3, \pm 1.5, \pm 10, \pm 5, \pm 2.5, \pm 6$

Step 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 1.3.1

Substitute $2$ into the polynomial.

$2 \cdot 2^{3} - 3 \cdot 2^{2} - 17 \cdot 2 + 30$

Step 1.3.2

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

$2 \cdot 8 - 3 \cdot 2^{2} - 17 \cdot 2 + 30$

Step 1.3.3

Multiply $2$ by $8$ .

$16 - 3 \cdot 2^{2} - 17 \cdot 2 + 30$

Step 1.3.4

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

$16 - 3 \cdot 4 - 17 \cdot 2 + 30$

Step 1.3.5

Multiply $- 3$ by $4$ .

$16 - 12 - 17 \cdot 2 + 30$

Step 1.3.6

Subtract $12$ from $16$ .

$4 - 17 \cdot 2 + 30$

Step 1.3.7

Multiply $- 17$ by $2$ .

$4 - 34 + 30$

Step 1.3.8

Subtract $34$ from $4$ .

$- 30 + 30$

Step 1.3.9

Add $- 30$ and $30$ .

$0$

Step 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.

$\frac{2 x^{3} - 3 x^{2} - 17 x + 30}{x - 2}$

Step 1.5

Divide $2 x^{3} - 3 x^{2} - 17 x + 30$ by $x - 2$ .

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Step 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 $0$ .

$x$

$2$

$2 x^{3}$

$3 x^{2}$

$17 x$

$30$

Step 1.5.2

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

					$2 x^{2}$
	$x$	-	$2$		$2 x^{3}$	-	$3 x^{2}$	-	$17 x$	+	$30$

Step 1.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	-	$3 x^{2}$	-	$17 x$	+	$30$
			+	$2 x^{3}$	-	$4 x^{2}$

Step 1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{3} - 4 x^{2}$

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	-	$3 x^{2}$	-	$17 x$	+	$30$
			-	$2 x^{3}$	+	$4 x^{2}$

Step 1.5.5

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

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	-	$3 x^{2}$	-	$17 x$	+	$30$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$x^{2}$

Step 1.5.6

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

				$2 x^{2}$
$x$	-	$2$		$2 x^{3}$	-	$3 x^{2}$	-	$17 x$	+	$30$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$x^{2}$	-	$17 x$

Step 1.5.7

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

				$2 x^{2}$	+	$x$
$x$	-	$2$		$2 x^{3}$	-	$3 x^{2}$	-	$17 x$	+	$30$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$x^{2}$	-	$17 x$

Step 1.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$x$
$x$	-	$2$		$2 x^{3}$	-	$3 x^{2}$	-	$17 x$	+	$30$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$x^{2}$	-	$17 x$
					+	$x^{2}$	-	$2 x$

Step 1.5.9

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

				$2 x^{2}$	+	$x$
$x$	-	$2$		$2 x^{3}$	-	$3 x^{2}$	-	$17 x$	+	$30$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$x^{2}$	-	$17 x$
					-	$x^{2}$	+	$2 x$

Step 1.5.10

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

				$2 x^{2}$	+	$x$
$x$	-	$2$		$2 x^{3}$	-	$3 x^{2}$	-	$17 x$	+	$30$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$x^{2}$	-	$17 x$
					-	$x^{2}$	+	$2 x$
							-	$15 x$

Step 1.5.11

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

				$2 x^{2}$	+	$x$
$x$	-	$2$		$2 x^{3}$	-	$3 x^{2}$	-	$17 x$	+	$30$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$x^{2}$	-	$17 x$
					-	$x^{2}$	+	$2 x$
							-	$15 x$	+	$30$

Step 1.5.12

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

				$2 x^{2}$	+	$x$	-	$15$
$x$	-	$2$		$2 x^{3}$	-	$3 x^{2}$	-	$17 x$	+	$30$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$x^{2}$	-	$17 x$
					-	$x^{2}$	+	$2 x$
							-	$15 x$	+	$30$

Step 1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$x$	-	$15$
$x$	-	$2$		$2 x^{3}$	-	$3 x^{2}$	-	$17 x$	+	$30$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$x^{2}$	-	$17 x$
					-	$x^{2}$	+	$2 x$
							-	$15 x$	+	$30$
							-	$15 x$	+	$30$

Step 1.5.14

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

				$2 x^{2}$	+	$x$	-	$15$
$x$	-	$2$		$2 x^{3}$	-	$3 x^{2}$	-	$17 x$	+	$30$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$x^{2}$	-	$17 x$
					-	$x^{2}$	+	$2 x$
							-	$15 x$	+	$30$
							+	$15 x$	-	$30$

Step 1.5.15

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

				$2 x^{2}$	+	$x$	-	$15$
$x$	-	$2$		$2 x^{3}$	-	$3 x^{2}$	-	$17 x$	+	$30$
			-	$2 x^{3}$	+	$4 x^{2}$
					+	$x^{2}$	-	$17 x$
					-	$x^{2}$	+	$2 x$
							-	$15 x$	+	$30$
							+	$15 x$	-	$30$
										$0$

Step 1.5.16

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

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

Step 1.6

Write $2 x^{3} - 3 x^{2} - 17 x + 30$ as a set of factors.

$(x - 2) (2 x^{2} + x - 15)$

Step 2

Factor by grouping.

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Step 2.1

Factor by grouping.

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Step 2.1.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 15 = - 30$ and whose sum is $b = 1$ .

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Step 2.1.1.1

Multiply by $1$ .

$(x - 2) (2 x^{2} + 1 x - 15)$

Step 2.1.1.2

Rewrite $1$ as $- 5$ plus $6$

$(x - 2) (2 x^{2} + (- 5 + 6) x - 15)$

Step 2.1.1.3

Apply the distributive property.

$(x - 2) (2 x^{2} - 5 x + 6 x - 15)$

Step 2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1

Group the first two terms and the last two terms.

$(x - 2) ((2 x^{2} - 5 x) + 6 x - 15)$

Step 2.1.2.2

Factor out the greatest common factor (GCF) from each group.

$(x - 2) (x (2 x - 5) + 3 (2 x - 5))$

Step 2.1.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 5$ .

$(x - 2) ((2 x - 5) (x + 3))$

Step 2.2

Remove unnecessary parentheses.

$(x - 2) (2 x - 5) (x + 3)$