Precalculus Examples

$f (x) = - x^{2} (x - 2) (x + 5)$

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $- x^{2} (x - 2) (x + 5) = 0$ .

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

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 $0$ .

$x^{2} = 0$

$x - 2 = 0$

$x + 5 = 0$

Step 1.2.3

Set $x^{2}$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $x^{2}$ equal to $0$ .

$x^{2} = 0$

Step 1.2.3.2

Solve $x^{2} = 0$ 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.

$x = \pm \sqrt{0}$

Step 1.2.3.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.3.2.2.1

Rewrite $0$ as $0^{2}$ .

$x = \pm \sqrt{0^{2}}$

Step 1.2.3.2.2.2

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

$x = \pm 0$

Step 1.2.3.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.2.4

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

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Step 1.2.4.1

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

$x - 2 = 0$

Step 1.2.4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 1.2.5

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

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Step 1.2.5.1

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

$x + 5 = 0$

Step 1.2.5.2

Subtract $5$ from both sides of the equation.

$x = - 5$

Step 1.2.6

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

$x = 0, 2, - 5$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (2, 0), (- 5, 0)$

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 $y$ .

$y = - {(0)}^{2} ((0) - 2) ((0) + 5)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = - 0^{2} ((0) - 2) ((0) + 5)$

Step 2.2.2

Remove parentheses.

$y = - 0^{2} (0 - 2) ((0) + 5)$

Step 2.2.3

Remove parentheses.

$y = - 0^{2} (0 - 2) (0 + 5)$

Step 2.2.4

Remove parentheses.

$y = - {(0)}^{2} ((0) - 2) ((0) + 5)$

Step 2.2.5

Simplify $- {(0)}^{2} ((0) - 2) ((0) + 5)$ .

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Step 2.2.5.1

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

$y = - 0 ((0) - 2) ((0) + 5)$

Step 2.2.5.2

Multiply $- 1$ by $0$ .

$y = 0 ((0) - 2) ((0) + 5)$

Step 2.2.5.3

Subtract $2$ from $0$ .

$y = 0 \cdot - 2 ((0) + 5)$

Step 2.2.5.4

Multiply $0$ by $- 2$ .

$y = 0 ((0) + 5)$

Step 2.2.5.5

Add $0$ and $5$ .

$y = 0 \cdot 5$

Step 2.2.5.6

Multiply $0$ by $5$ .

$y = 0$

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (2, 0), (- 5, 0)$

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

Step 4