Precalculus Examples

$f (x) = - x^{3} {(3 x - 4)}^{2} (x + 2)$

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 $x$ .

$0 = - x^{3} {(3 x - 4)}^{2} (x + 2)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- x^{3} {(3 x - 4)}^{2} (x + 2) = 0$ .

$- x^{3} {(3 x - 4)}^{2} (x + 2) = 0$

Step 1.2.2

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

$x^{3} = 0$

${(3 x - 4)}^{2} = 0$

$x + 2 = 0$

Step 1.2.3

Set $x^{3}$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $x^{3}$ equal to $0$ .

$x^{3} = 0$

Step 1.2.3.2

Solve $x^{3} = 0$ for $x$ .

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Step 1.2.3.2.1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \sqrt[3]{0}$

Step 1.2.3.2.2

Simplify $\sqrt[3]{0}$ .

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Step 1.2.3.2.2.1

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

Step 1.2.3.2.2.2

Pull terms out from under the radical, assuming real numbers.

$x = 0$

Step 1.2.4

Set ${(3 x - 4)}^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set ${(3 x - 4)}^{2}$ equal to $0$ .

${(3 x - 4)}^{2} = 0$

Step 1.2.4.2

Solve ${(3 x - 4)}^{2} = 0$ for $x$ .

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Step 1.2.4.2.1

Set the $3 x - 4$ equal to $0$ .

$3 x - 4 = 0$

Step 1.2.4.2.2

Solve for $x$ .

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Step 1.2.4.2.2.1

Add $4$ to both sides of the equation.

$3 x = 4$

Step 1.2.4.2.2.2

Divide each term in $3 x = 4$ by $3$ and simplify.

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Step 1.2.4.2.2.2.1

Divide each term in $3 x = 4$ by $3$ .

$\frac{3 x}{3} = \frac{4}{3}$

Step 1.2.4.2.2.2.2

Simplify the left side.

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Step 1.2.4.2.2.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.4.2.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{4}{3}$

Step 1.2.4.2.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{4}{3}$

Step 1.2.5

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

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Step 1.2.5.1

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

$x + 2 = 0$

Step 1.2.5.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 1.2.6

The final solution is all the values that make $- x^{3} {(3 x - 4)}^{2} (x + 2) = 0$ true.

$x = 0, \frac{4}{3}, - 2$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (\frac{4}{3}, 0), (- 2, 0)$

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 $y$ .

$y = - {(0)}^{3} {(3 (0) - 4)}^{2} ((0) + 2)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 0^{3} {(3 (0) - 4)}^{2} ((0) + 2)$

Step 2.2.2

Remove parentheses.

$y = - 0^{3} {(3 (0) - 4)}^{2} (0 + 2)$

Step 2.2.3

Remove parentheses.

$y = - {(0)}^{3} {(3 (0) - 4)}^{2} ((0) + 2)$

Step 2.2.4

Simplify $- {(0)}^{3} {(3 (0) - 4)}^{2} ((0) + 2)$ .

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Step 2.2.4.1

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

$y = - 0 {(3 (0) - 4)}^{2} ((0) + 2)$

Step 2.2.4.2

Multiply $- 1$ by $0$ .

$y = 0 {(3 (0) - 4)}^{2} ((0) + 2)$

Step 2.2.4.3

Multiply $3$ by $0$ .

$y = 0 {(0 - 4)}^{2} ((0) + 2)$

Step 2.2.4.4

Subtract $4$ from $0$ .

$y = 0 {(- 4)}^{2} ((0) + 2)$

Step 2.2.4.5

Raise $- 4$ to the power of $2$ .

$y = 0 \cdot 16 ((0) + 2)$

Step 2.2.4.6

Multiply $0$ by $16$ .

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

Step 2.2.4.7

Add $0$ and $2$ .

$y = 0 \cdot 2$

Step 2.2.4.8

Multiply $0$ by $2$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (\frac{4}{3}, 0), (- 2, 0)$

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

Step 4