Precalculus Examples

$f (x) = 5 x^{3} + 12 x^{2} + 9 x + 2$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = 5 x^{3} + 12 x^{2} + 9 x + 2$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $5 x^{3} + 12 x^{2} + 9 x + 2 = 0$ .

$5 x^{3} + 12 x^{2} + 9 x + 2 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $5 x^{3} + 12 x^{2} + 9 x + 2$ using the rational roots test.

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Step 1.2.2.1.1

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.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 5$

Step 1.2.2.1.2

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

$\pm 1, \pm 0.2, \pm 2, \pm 0.4$

Step 1.2.2.1.3

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 1.2.2.1.3.1

Substitute $- 1$ into the polynomial.

$5 {(- 1)}^{3} + 12 {(- 1)}^{2} + 9 \cdot - 1 + 2$

Step 1.2.2.1.3.2

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

$5 \cdot - 1 + 12 {(- 1)}^{2} + 9 \cdot - 1 + 2$

Step 1.2.2.1.3.3

Multiply $5$ by $- 1$ .

$- 5 + 12 {(- 1)}^{2} + 9 \cdot - 1 + 2$

Step 1.2.2.1.3.4

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

$- 5 + 12 \cdot 1 + 9 \cdot - 1 + 2$

Step 1.2.2.1.3.5

Multiply $12$ by $1$ .

$- 5 + 12 + 9 \cdot - 1 + 2$

Step 1.2.2.1.3.6

Add $- 5$ and $12$ .

$7 + 9 \cdot - 1 + 2$

Step 1.2.2.1.3.7

Multiply $9$ by $- 1$ .

$7 - 9 + 2$

Step 1.2.2.1.3.8

Subtract $9$ from $7$ .

$- 2 + 2$

Step 1.2.2.1.3.9

Add $- 2$ and $2$ .

$0$

Step 1.2.2.1.4

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.

$\frac{5 x^{3} + 12 x^{2} + 9 x + 2}{x + 1}$

Step 1.2.2.1.5

Divide $5 x^{3} + 12 x^{2} + 9 x + 2$ by $x + 1$ .

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Step 1.2.2.1.5.1

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

$x$

$1$

$5 x^{3}$

$12 x^{2}$

$9 x$

$2$

Step 1.2.2.1.5.2

Divide the highest order term in the dividend $5 x^{3}$ by the highest order term in divisor $x$ .

					$5 x^{2}$
	$x$	+	$1$		$5 x^{3}$	+	$12 x^{2}$	+	$9 x$	+	$2$

Step 1.2.2.1.5.3

Multiply the new quotient term by the divisor.

				$5 x^{2}$
$x$	+	$1$		$5 x^{3}$	+	$12 x^{2}$	+	$9 x$	+	$2$
			+	$5 x^{3}$	+	$5 x^{2}$

Step 1.2.2.1.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $5 x^{3} + 5 x^{2}$

				$5 x^{2}$
$x$	+	$1$		$5 x^{3}$	+	$12 x^{2}$	+	$9 x$	+	$2$
			-	$5 x^{3}$	-	$5 x^{2}$

Step 1.2.2.1.5.5

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

				$5 x^{2}$
$x$	+	$1$		$5 x^{3}$	+	$12 x^{2}$	+	$9 x$	+	$2$
			-	$5 x^{3}$	-	$5 x^{2}$
					+	$7 x^{2}$

Step 1.2.2.1.5.6

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

				$5 x^{2}$
$x$	+	$1$		$5 x^{3}$	+	$12 x^{2}$	+	$9 x$	+	$2$
			-	$5 x^{3}$	-	$5 x^{2}$
					+	$7 x^{2}$	+	$9 x$

Step 1.2.2.1.5.7

Divide the highest order term in the dividend $7 x^{2}$ by the highest order term in divisor $x$ .

				$5 x^{2}$	+	$7 x$
$x$	+	$1$		$5 x^{3}$	+	$12 x^{2}$	+	$9 x$	+	$2$
			-	$5 x^{3}$	-	$5 x^{2}$
					+	$7 x^{2}$	+	$9 x$

Step 1.2.2.1.5.8

Multiply the new quotient term by the divisor.

				$5 x^{2}$	+	$7 x$
$x$	+	$1$		$5 x^{3}$	+	$12 x^{2}$	+	$9 x$	+	$2$
			-	$5 x^{3}$	-	$5 x^{2}$
					+	$7 x^{2}$	+	$9 x$
					+	$7 x^{2}$	+	$7 x$

Step 1.2.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{2} + 7 x$

				$5 x^{2}$	+	$7 x$
$x$	+	$1$		$5 x^{3}$	+	$12 x^{2}$	+	$9 x$	+	$2$
			-	$5 x^{3}$	-	$5 x^{2}$
					+	$7 x^{2}$	+	$9 x$
					-	$7 x^{2}$	-	$7 x$

Step 1.2.2.1.5.10

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

				$5 x^{2}$	+	$7 x$
$x$	+	$1$		$5 x^{3}$	+	$12 x^{2}$	+	$9 x$	+	$2$
			-	$5 x^{3}$	-	$5 x^{2}$
					+	$7 x^{2}$	+	$9 x$
					-	$7 x^{2}$	-	$7 x$
							+	$2 x$

Step 1.2.2.1.5.11

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

				$5 x^{2}$	+	$7 x$
$x$	+	$1$		$5 x^{3}$	+	$12 x^{2}$	+	$9 x$	+	$2$
			-	$5 x^{3}$	-	$5 x^{2}$
					+	$7 x^{2}$	+	$9 x$
					-	$7 x^{2}$	-	$7 x$
							+	$2 x$	+	$2$

Step 1.2.2.1.5.12

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

				$5 x^{2}$	+	$7 x$	+	$2$
$x$	+	$1$		$5 x^{3}$	+	$12 x^{2}$	+	$9 x$	+	$2$
			-	$5 x^{3}$	-	$5 x^{2}$
					+	$7 x^{2}$	+	$9 x$
					-	$7 x^{2}$	-	$7 x$
							+	$2 x$	+	$2$

Step 1.2.2.1.5.13

Multiply the new quotient term by the divisor.

				$5 x^{2}$	+	$7 x$	+	$2$
$x$	+	$1$		$5 x^{3}$	+	$12 x^{2}$	+	$9 x$	+	$2$
			-	$5 x^{3}$	-	$5 x^{2}$
					+	$7 x^{2}$	+	$9 x$
					-	$7 x^{2}$	-	$7 x$
							+	$2 x$	+	$2$
							+	$2 x$	+	$2$

Step 1.2.2.1.5.14

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

				$5 x^{2}$	+	$7 x$	+	$2$
$x$	+	$1$		$5 x^{3}$	+	$12 x^{2}$	+	$9 x$	+	$2$
			-	$5 x^{3}$	-	$5 x^{2}$
					+	$7 x^{2}$	+	$9 x$
					-	$7 x^{2}$	-	$7 x$
							+	$2 x$	+	$2$
							-	$2 x$	-	$2$

Step 1.2.2.1.5.15

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

				$5 x^{2}$	+	$7 x$	+	$2$
$x$	+	$1$		$5 x^{3}$	+	$12 x^{2}$	+	$9 x$	+	$2$
			-	$5 x^{3}$	-	$5 x^{2}$
					+	$7 x^{2}$	+	$9 x$
					-	$7 x^{2}$	-	$7 x$
							+	$2 x$	+	$2$
							-	$2 x$	-	$2$
										$0$

Step 1.2.2.1.5.16

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

$5 x^{2} + 7 x + 2$

Step 1.2.2.1.6

Write $5 x^{3} + 12 x^{2} + 9 x + 2$ as a set of factors.

$(x + 1) (5 x^{2} + 7 x + 2) = 0$

Step 1.2.2.2

Factor by grouping.

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Step 1.2.2.2.1

Factor by grouping.

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Step 1.2.2.2.1.1

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 $a \cdot c = 5 \cdot 2 = 10$ and whose sum is $b = 7$ .

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Step 1.2.2.2.1.1.1

Factor $7$ out of $7 x$ .

$(x + 1) (5 x^{2} + 7 (x) + 2) = 0$

Step 1.2.2.2.1.1.2

Rewrite $7$ as $2$ plus $5$

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

Step 1.2.2.2.1.1.3

Apply the distributive property.

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

Step 1.2.2.2.1.2

Factor out the greatest common factor from each group.

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Step 1.2.2.2.1.2.1

Group the first two terms and the last two terms.

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

Step 1.2.2.2.1.2.2

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

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

Step 1.2.2.2.1.3

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

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

Step 1.2.2.2.2

Remove unnecessary parentheses.

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

Step 1.2.3

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

$x + 1 = 0$

$5 x + 2 = 0$

$x + 1 = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

$x + 1 = 0$

Step 1.2.4.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 1.2.5

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

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Step 1.2.5.1

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

$5 x + 2 = 0$

Step 1.2.5.2

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

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Step 1.2.5.2.1

Subtract $2$ from both sides of the equation.

$5 x = - 2$

Step 1.2.5.2.2

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

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Step 1.2.5.2.2.1

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

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

Step 1.2.5.2.2.2

Simplify the left side.

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Step 1.2.5.2.2.2.1

Cancel the common factor of $5$ .

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Step 1.2.5.2.2.2.1.1

Cancel the common factor.

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

Step 1.2.5.2.2.2.1.2

Divide $x$ by $1$ .

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

Step 1.2.5.2.2.3

Simplify the right side.

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Step 1.2.5.2.2.3.1

Move the negative in front of the fraction.

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

Step 1.2.6

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

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

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 1, 0), (- \frac{2}{5}, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = 5 {(0)}^{3} + 12 {(0)}^{2} + 9 (0) + 2$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = 5 \cdot 0^{3} + 12 {(0)}^{2} + 9 (0) + 2$

Step 2.2.2

Remove parentheses.

$y = 5 \cdot 0^{3} + 12 \cdot 0^{2} + 9 (0) + 2$

Step 2.2.3

Remove parentheses.

$y = 5 {(0)}^{3} + 12 {(0)}^{2} + 9 (0) + 2$

Step 2.2.4

Simplify $5 {(0)}^{3} + 12 {(0)}^{2} + 9 (0) + 2$ .

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Step 2.2.4.1

Simplify each term.

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Step 2.2.4.1.1

Raising $0$ to any positive power yields $0$ .

$y = 5 \cdot 0 + 12 {(0)}^{2} + 9 (0) + 2$

Step 2.2.4.1.2

Multiply $5$ by $0$ .

$y = 0 + 12 {(0)}^{2} + 9 (0) + 2$

Step 2.2.4.1.3

Raising $0$ to any positive power yields $0$ .

$y = 0 + 12 \cdot 0 + 9 (0) + 2$

Step 2.2.4.1.4

Multiply $12$ by $0$ .

$y = 0 + 0 + 9 (0) + 2$

Step 2.2.4.1.5

Multiply $9$ by $0$ .

$y = 0 + 0 + 0 + 2$

Step 2.2.4.2

Simplify by adding numbers.

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Step 2.2.4.2.1

Add $0$ and $0$ .

$y = 0 + 0 + 2$

Step 2.2.4.2.2

Add $0$ and $0$ .

$y = 0 + 2$

Step 2.2.4.2.3

Add $0$ and $2$ .

$y = 2$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 2)$

Step 3

List the intersections.

x-intercept(s): $(- 1, 0), (- \frac{2}{5}, 0)$

y-intercept(s): $(0, 2)$

Step 4