Precalculus Examples

$x^{3} - x$

Step 1

Write $x^{3} - x$ as an equation.

$y = x^{3} - x$

Step 2

Find the x-intercepts.

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Step 2.1

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

$0 = x^{3} - x$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $x^{3} - x = 0$ .

$x^{3} - x = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Factor $x$ out of $x^{3} - x$ .

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Step 2.2.2.1.1

Factor $x$ out of $x^{3}$ .

$x \cdot x^{2} - x = 0$

Step 2.2.2.1.2

Factor $x$ out of $- x$ .

$x \cdot x^{2} + x \cdot - 1 = 0$

Step 2.2.2.1.3

Factor $x$ out of $x \cdot x^{2} + x \cdot - 1$ .

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

Step 2.2.2.2

Rewrite $1$ as $1^{2}$ .

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

Step 2.2.2.3

Factor.

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Step 2.2.2.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$x ((x + 1) (x - 1)) = 0$

Step 2.2.2.3.2

Remove unnecessary parentheses.

$x (x + 1) (x - 1) = 0$

Step 2.2.3

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

$x = 0$

$x + 1 = 0$

$x - 1 = 0$

Step 2.2.4

Set $x$ equal to $0$ .

$x = 0$

Step 2.2.5

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

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Step 2.2.5.1

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

$x + 1 = 0$

Step 2.2.5.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.2.6

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

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Step 2.2.6.1

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

$x - 1 = 0$

Step 2.2.6.2

Add $1$ to both sides of the equation.

$x = 1$

Step 2.2.7

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

$x = 0, - 1, 1$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 1, 0), (1, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

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

$y = {(0)}^{3} - (0)$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = 0^{3} - (0)$

Step 3.2.2

Remove parentheses.

$y = {(0)}^{3} - (0)$

Step 3.2.3

Simplify ${(0)}^{3} - (0)$ .

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Step 3.2.3.1

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

$y = 0 - (0)$

Step 3.2.3.2

Subtract $0$ from $0$ .

$y = 0$

Step 3.3

y-intercept(s) in point form.

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

Step 4

List the intersections.

x-intercept(s): $(0, 0), (- 1, 0), (1, 0)$

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

Step 5