Precalculus Examples

$f (x) = 15 x^{3} + 14 x^{2} - 3 x - 2$

Step 1

Set $15 x^{3} + 14 x^{2} - 3 x - 2$ equal to $0$ .

$15 x^{3} + 14 x^{2} - 3 x - 2 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $15 x^{3} + 14 x^{2} - 3 x - 2$ using the rational roots test.

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Step 2.1.1.1

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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 15, \pm 3, \pm 5$

Step 2.1.1.2

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

$\pm 1, \pm 0.0\overline{6}, \pm 0.\overline{3}, \pm 0.2, \pm 2, \pm 0.1\overline{3}, \pm 0.\overline{6}, \pm 0.4$

Step 2.1.1.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 2.1.1.3.1

Substitute $- 1$ into the polynomial.

$15 {(- 1)}^{3} + 14 {(- 1)}^{2} - 3 \cdot - 1 - 2$

Step 2.1.1.3.2

Raise $- 1$ to the power of $3$ .

$15 \cdot - 1 + 14 {(- 1)}^{2} - 3 \cdot - 1 - 2$

Step 2.1.1.3.3

Multiply $15$ by $- 1$ .

$- 15 + 14 {(- 1)}^{2} - 3 \cdot - 1 - 2$

Step 2.1.1.3.4

Raise $- 1$ to the power of $2$ .

$- 15 + 14 \cdot 1 - 3 \cdot - 1 - 2$

Step 2.1.1.3.5

Multiply $14$ by $1$ .

$- 15 + 14 - 3 \cdot - 1 - 2$

Step 2.1.1.3.6

Add $- 15$ and $14$ .

$- 1 - 3 \cdot - 1 - 2$

Step 2.1.1.3.7

Multiply $- 3$ by $- 1$ .

$- 1 + 3 - 2$

Step 2.1.1.3.8

Add $- 1$ and $3$ .

$2 - 2$

Step 2.1.1.3.9

Subtract $2$ from $2$ .

$0$

Step 2.1.1.4

Since $- 1$ is a known root, divide the polynomial by $x + 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{15 x^{3} + 14 x^{2} - 3 x - 2}{x + 1}$

Step 2.1.1.5

Divide $15 x^{3} + 14 x^{2} - 3 x - 2$ by $x + 1$ .

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Step 2.1.1.5.1

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

$x$

$1$

$15 x^{3}$

$14 x^{2}$

$3 x$

$2$

Step 2.1.1.5.2

Divide the highest order term in the dividend $15 x^{3}$ by the highest order term in divisor $x$ .

					$15 x^{2}$
	$x$	+	$1$		$15 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$2$

Step 2.1.1.5.3

Multiply the new quotient term by the divisor.

				$15 x^{2}$
$x$	+	$1$		$15 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$2$
			+	$15 x^{3}$	+	$15 x^{2}$

Step 2.1.1.5.4

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

				$15 x^{2}$
$x$	+	$1$		$15 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$2$
			-	$15 x^{3}$	-	$15 x^{2}$

Step 2.1.1.5.5

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

				$15 x^{2}$
$x$	+	$1$		$15 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$2$
			-	$15 x^{3}$	-	$15 x^{2}$
					-	$x^{2}$

Step 2.1.1.5.6

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

				$15 x^{2}$
$x$	+	$1$		$15 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$2$
			-	$15 x^{3}$	-	$15 x^{2}$
					-	$x^{2}$	-	$3 x$

Step 2.1.1.5.7

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

				$15 x^{2}$	-	$x$
$x$	+	$1$		$15 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$2$
			-	$15 x^{3}$	-	$15 x^{2}$
					-	$x^{2}$	-	$3 x$

Step 2.1.1.5.8

Multiply the new quotient term by the divisor.

				$15 x^{2}$	-	$x$
$x$	+	$1$		$15 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$2$
			-	$15 x^{3}$	-	$15 x^{2}$
					-	$x^{2}$	-	$3 x$
					-	$x^{2}$	-	$x$

Step 2.1.1.5.9

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

				$15 x^{2}$	-	$x$
$x$	+	$1$		$15 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$2$
			-	$15 x^{3}$	-	$15 x^{2}$
					-	$x^{2}$	-	$3 x$
					+	$x^{2}$	+	$x$

Step 2.1.1.5.10

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

				$15 x^{2}$	-	$x$
$x$	+	$1$		$15 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$2$
			-	$15 x^{3}$	-	$15 x^{2}$
					-	$x^{2}$	-	$3 x$
					+	$x^{2}$	+	$x$
							-	$2 x$

Step 2.1.1.5.11

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

				$15 x^{2}$	-	$x$
$x$	+	$1$		$15 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$2$
			-	$15 x^{3}$	-	$15 x^{2}$
					-	$x^{2}$	-	$3 x$
					+	$x^{2}$	+	$x$
							-	$2 x$	-	$2$

Step 2.1.1.5.12

Divide the highest order term in the dividend $- 2 x$ by the highest order term in divisor $x$ .

				$15 x^{2}$	-	$x$	-	$2$
$x$	+	$1$		$15 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$2$
			-	$15 x^{3}$	-	$15 x^{2}$
					-	$x^{2}$	-	$3 x$
					+	$x^{2}$	+	$x$
							-	$2 x$	-	$2$

Step 2.1.1.5.13

Multiply the new quotient term by the divisor.

				$15 x^{2}$	-	$x$	-	$2$
$x$	+	$1$		$15 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$2$
			-	$15 x^{3}$	-	$15 x^{2}$
					-	$x^{2}$	-	$3 x$
					+	$x^{2}$	+	$x$
							-	$2 x$	-	$2$
							-	$2 x$	-	$2$

Step 2.1.1.5.14

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

				$15 x^{2}$	-	$x$	-	$2$
$x$	+	$1$		$15 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$2$
			-	$15 x^{3}$	-	$15 x^{2}$
					-	$x^{2}$	-	$3 x$
					+	$x^{2}$	+	$x$
							-	$2 x$	-	$2$
							+	$2 x$	+	$2$

Step 2.1.1.5.15

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

				$15 x^{2}$	-	$x$	-	$2$
$x$	+	$1$		$15 x^{3}$	+	$14 x^{2}$	-	$3 x$	-	$2$
			-	$15 x^{3}$	-	$15 x^{2}$
					-	$x^{2}$	-	$3 x$
					+	$x^{2}$	+	$x$
							-	$2 x$	-	$2$
							+	$2 x$	+	$2$
										$0$

Step 2.1.1.5.16

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

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

Step 2.1.1.6

Write $15 x^{3} + 14 x^{2} - 3 x - 2$ as a set of factors.

$(x + 1) (15 x^{2} - x - 2) = 0$

Step 2.1.2

Factor by grouping.

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Step 2.1.2.1

Factor by grouping.

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Step 2.1.2.1.1

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 $a \cdot c = 15 \cdot - 2 = - 30$ and whose sum is $b = - 1$ .

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Step 2.1.2.1.1.1

Factor $- 1$ out of $- x$ .

$(x + 1) (15 x^{2} - (x) - 2) = 0$

Step 2.1.2.1.1.2

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

$(x + 1) (15 x^{2} + (5 - 6) x - 2) = 0$

Step 2.1.2.1.1.3

Apply the distributive property.

$(x + 1) (15 x^{2} + 5 x - 6 x - 2) = 0$

Step 2.1.2.1.2

Factor out the greatest common factor from each group.

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Step 2.1.2.1.2.1

Group the first two terms and the last two terms.

$(x + 1) ((15 x^{2} + 5 x) - 6 x - 2) = 0$

Step 2.1.2.1.2.2

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

$(x + 1) (5 x (3 x + 1) - 2 (3 x + 1)) = 0$

Step 2.1.2.1.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 1$ .

$(x + 1) ((3 x + 1) (5 x - 2)) = 0$

Step 2.1.2.2

Remove unnecessary parentheses.

$(x + 1) (3 x + 1) (5 x - 2) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 1 = 0$

$3 x + 1 = 0$

$5 x - 2 = 0$

Step 2.3

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.4

Set $3 x + 1$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $3 x + 1$ equal to $0$ .

$3 x + 1 = 0$

Step 2.4.2

Solve $3 x + 1 = 0$ for $x$ .

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Step 2.4.2.1

Subtract $1$ from both sides of the equation.

$3 x = - 1$

Step 2.4.2.2

Divide each term in $3 x = - 1$ by $3$ and simplify.

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Step 2.4.2.2.1

Divide each term in $3 x = - 1$ by $3$ .

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 2.4.2.2.2

Simplify the left side.

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Step 2.4.2.2.2.1

Cancel the common factor of $3$ .

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Step 2.4.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 1}{3}$

Step 2.4.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 1}{3}$

Step 2.4.2.2.3

Simplify the right side.

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Step 2.4.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{3}$

Step 2.5

Set $5 x - 2$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $5 x - 2$ equal to $0$ .

$5 x - 2 = 0$

Step 2.5.2

Solve $5 x - 2 = 0$ for $x$ .

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Step 2.5.2.1

Add $2$ to both sides of the equation.

$5 x = 2$

Step 2.5.2.2

Divide each term in $5 x = 2$ by $5$ and simplify.

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Step 2.5.2.2.1

Divide each term in $5 x = 2$ by $5$ .

$\frac{5 x}{5} = \frac{2}{5}$

Step 2.5.2.2.2

Simplify the left side.

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Step 2.5.2.2.2.1

Cancel the common factor of $5$ .

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Step 2.5.2.2.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{2}{5}$

Step 2.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{2}{5}$

Step 2.6

The final solution is all the values that make $(x + 1) (3 x + 1) (5 x - 2) = 0$ true.

$x = - 1, - \frac{1}{3}, \frac{2}{5}$

Step 3