Find the x and y Intercepts P(x)=4/9x^3-2x^2

$P (x) = \frac{4}{9} x^{3} - 2 x^{2}$

Step 1

Find the x-intercepts.

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

$0 = \frac{4}{9} x^{3} - 2 x^{2}$

Solve the equation.

Rewrite the equation as $\frac{4}{9} x^{3} - 2 x^{2} = 0$ .

$\frac{4}{9} x^{3} - 2 x^{2} = 0$

Combine $\frac{4}{9}$ and $x^{3}$ .

$\frac{4 x^{3}}{9} - 2 x^{2} = 0$

Multiply each term in $\frac{4 x^{3}}{9} - 2 x^{2} = 0$ by $9$ to eliminate the fractions.

Multiply each term in $\frac{4 x^{3}}{9} - 2 x^{2} = 0$ by $9$ .

$\frac{4 x^{3}}{9} \cdot 9 - 2 x^{2} \cdot 9 = 0 \cdot 9$

Simplify the left side.

Simplify each term.

Cancel the common factor of $9$ .

Cancel the common factor.

$\frac{4 x^{3}}{9} \cdot 9 - 2 x^{2} \cdot 9 = 0 \cdot 9$

Rewrite the expression.

$4 x^{3} - 2 x^{2} \cdot 9 = 0 \cdot 9$

Multiply $9$ by $- 2$ .

$4 x^{3} - 18 x^{2} = 0 \cdot 9$

Simplify the right side.

Multiply $0$ by $9$ .

$4 x^{3} - 18 x^{2} = 0$

Factor $2 x^{2}$ out of $4 x^{3} - 18 x^{2}$ .

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

$2 x^{2} (2 x) - 18 x^{2} = 0$

Factor $2 x^{2}$ out of $- 18 x^{2}$ .

$2 x^{2} (2 x) + 2 x^{2} (- 9) = 0$

Factor $2 x^{2}$ out of $2 x^{2} (2 x) + 2 x^{2} (- 9)$ .

$2 x^{2} (2 x - 9) = 0$

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

$x^{2} = 0$

$2 x - 9 = 0$

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

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

$x^{2} = 0$

Solve $x^{2} = 0$ for $x$ .

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

$x = \pm \sqrt{0}$

Simplify $\pm \sqrt{0}$ .

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

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

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

$x = \pm 0$

Plus or minus $0$ is $0$ .

$x = 0$

Set $2 x - 9$ equal to $0$ and solve for $x$ .

Set $2 x - 9$ equal to $0$ .

$2 x - 9 = 0$

Solve $2 x - 9 = 0$ for $x$ .

Add $9$ to both sides of the equation.

$2 x = 9$

Divide each term in $2 x = 9$ by $2$ and simplify.

Divide each term in $2 x = 9$ by $2$ .

$\frac{2 x}{2} = \frac{9}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{9}{2}$

Divide $x$ by $1$ .

$x = \frac{9}{2}$

The final solution is all the values that make $2 x^{2} (2 x - 9) = 0$ true.

$x = 0, \frac{9}{2}$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (\frac{9}{2}, 0)$

Step 2

Find the y-intercepts.

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

$y = \frac{4}{9} \cdot {(0)}^{3} - 2 {(0)}^{2}$

Solve the equation.

Remove parentheses.

$y = \frac{4}{9} \cdot 0^{3} - 2 {(0)}^{2}$

Remove parentheses.

$y = \frac{4}{9} \cdot 0^{3} - 2 \cdot 0^{2}$

Remove parentheses.

$y = \frac{4}{9} \cdot {(0)}^{3} - 2 {(0)}^{2}$

Simplify $\frac{4}{9} \cdot {(0)}^{3} - 2 {(0)}^{2}$ .

Simplify each term.

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

$y = \frac{4}{9} \cdot 0 - 2 {(0)}^{2}$

Multiply $\frac{4}{9}$ by $0$ .

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

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

$y = 0 - 2 \cdot 0$

Multiply $- 2$ by $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

y-intercept(s) in point form.

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

Step 3

List the intersections.

x-intercept(s): $(0, 0), (\frac{9}{2}, 0)$

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

Step 4

Find the x and y Intercepts P(x)=4/9x^3-2x^2

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