Algebra Examples

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

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 $x$ .

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

Step 1.2

Solve the equation.

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Step 1.2.1

Set the numerator equal to zero.

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

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

Subtract $4$ from both sides of the equation.

$- 2 x^{3} = - 4$

Step 1.2.2.2

Add $4$ to both sides of the equation.

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

Step 1.2.2.3

Factor $- 2$ out of $- 2 x^{3} + 4$ .

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Step 1.2.2.3.1

Factor $- 2$ out of $- 2 x^{3}$ .

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

Step 1.2.2.3.2

Factor $- 2$ out of $4$ .

$- 2 x^{3} - 2 \cdot - 2 = 0$

Step 1.2.2.3.3

Factor $- 2$ out of $- 2 (x^{3}) - 2 (- 2)$ .

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

Step 1.2.2.4

Divide each term in $- 2 (x^{3} - 2) = 0$ by $- 2$ and simplify.

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Step 1.2.2.4.1

Divide each term in $- 2 (x^{3} - 2) = 0$ by $- 2$ .

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

Step 1.2.2.4.2

Simplify the left side.

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Step 1.2.2.4.2.1

Cancel the common factor of $- 2$ .

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Step 1.2.2.4.2.1.1

Cancel the common factor.

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

Step 1.2.2.4.2.1.2

Divide $x^{3} - 2$ by $1$ .

$x^{3} - 2 = \frac{0}{- 2}$

Step 1.2.2.4.3

Simplify the right side.

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Step 1.2.2.4.3.1

Divide $0$ by $- 2$ .

$x^{3} - 2 = 0$

Step 1.2.2.5

Add $2$ to both sides of the equation.

$x^{3} = 2$

Step 1.2.2.6

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

$x = \sqrt[3]{2}$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(\sqrt[3]{2}, 0)$

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 $y$ .

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

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \frac{- 2 \cdot 0^{3} + 4}{3}$

Step 2.2.2

Remove parentheses.

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

Step 2.2.3

Simplify the numerator.

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Step 2.2.3.1

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

$y = \frac{- 2 \cdot 0 + 4}{3}$

Step 2.2.3.2

Multiply $- 2$ by $0$ .

$y = \frac{0 + 4}{3}$

Step 2.2.3.3

Add $0$ and $4$ .

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

Step 2.3

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(\sqrt[3]{2}, 0)$

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

Step 4