Find the x and y Intercepts f(x)=x-x^3

$f (x) = x - x^{3}$

Find the x-intercepts.

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

$0 = x - x^{3}$

Solve the equation.

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

$x - x^{3} = 0$

Factor the left side of the equation.

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

Raise $x$ to the power of $1$ .

$x - x^{3} = 0$

Factor $x$ out of $x^{1}$ .

$x \cdot 1 - x^{3} = 0$

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

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

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

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

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

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

Factor.

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

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

Remove unnecessary parentheses.

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

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

$x = 0$

$1 + x = 0$

$1 - x = 0$

Set $x$ equal to $0$ .

$x = 0$

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

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

$1 + x = 0$

Subtract $1$ from both sides of the equation.

$x = - 1$

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

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

$1 - x = 0$

Solve $1 - x = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$- x = - 1$

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

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

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

Simplify the left side.

Dividing two negative values results in a positive value.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Divide $- 1$ by $- 1$ .

$x = 1$

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

$x = 0, - 1, 1$

x-intercept(s) in point form.

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

Find the y-intercepts.

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

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

Solve the equation.

Remove parentheses.

$y = 0 - 0^{3}$

Remove parentheses.

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

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

Simplify each term.

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

$y = 0 - 0$

Multiply $- 1$ by $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

y-intercept(s) in point form.

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

List the intersections.

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

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

Find the x and y Intercepts f(x)=x-x^3

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