Find the x and y Intercepts f(x)=-3x^6-10x^5+15x^4+50x^3

$f (x) = - 3 x^{6} - 10 x^{5} + 15 x^{4} + 50 x^{3}$

Find the x-intercepts.

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

$0 = - 3 x^{6} - 10 x^{5} + 15 x^{4} + 50 x^{3}$

Solve the equation.

Rewrite the equation as $- 3 x^{6} - 10 x^{5} + 15 x^{4} + 50 x^{3} = 0$ .

$- 3 x^{6} - 10 x^{5} + 15 x^{4} + 50 x^{3} = 0$

Factor the left side of the equation.

Factor $x^{3}$ out of $- 3 x^{6} - 10 x^{5} + 15 x^{4} + 50 x^{3}$ .

Factor $x^{3}$ out of $- 3 x^{6}$ .

$x^{3} (- 3 x^{3}) - 10 x^{5} + 15 x^{4} + 50 x^{3} = 0$

Factor $x^{3}$ out of $- 10 x^{5}$ .

$x^{3} (- 3 x^{3}) + x^{3} (- 10 x^{2}) + 15 x^{4} + 50 x^{3} = 0$

Factor $x^{3}$ out of $15 x^{4}$ .

$x^{3} (- 3 x^{3}) + x^{3} (- 10 x^{2}) + x^{3} (15 x) + 50 x^{3} = 0$

Factor $x^{3}$ out of $50 x^{3}$ .

$x^{3} (- 3 x^{3}) + x^{3} (- 10 x^{2}) + x^{3} (15 x) + x^{3} \cdot 50 = 0$

Factor $x^{3}$ out of $x^{3} (- 3 x^{3}) + x^{3} (- 10 x^{2})$ .

$x^{3} (- 3 x^{3} - 10 x^{2}) + x^{3} (15 x) + x^{3} \cdot 50 = 0$

Factor $x^{3}$ out of $x^{3} (- 3 x^{3} - 10 x^{2}) + x^{3} (15 x)$ .

$x^{3} (- 3 x^{3} - 10 x^{2} + 15 x) + x^{3} \cdot 50 = 0$

Factor $x^{3}$ out of $x^{3} (- 3 x^{3} - 10 x^{2} + 15 x) + x^{3} \cdot 50$ .

$x^{3} (- 3 x^{3} - 10 x^{2} + 15 x + 50) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$x^{3} ((- 3 x^{3} - 10 x^{2}) + 15 x + 50) = 0$

Factor out the greatest common factor (GCF) from each group.

$x^{3} (x^{2} (- 3 x - 10) - 5 (- 3 x - 10)) = 0$

Factor.

Factor the polynomial by factoring out the greatest common factor, $- 3 x - 10$ .

$x^{3} ((- 3 x - 10) (x^{2} - 5)) = 0$

Remove unnecessary parentheses.

$x^{3} (- 3 x - 10) (x^{2} - 5) = 0$

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

$x^{3} = 0$

$- 3 x - 10 = 0$

$x^{2} - 5 = 0$

Set $x^{3}$ equal to $0$ and solve for $x$ .

Set $x^{3}$ equal to $0$ .

$x^{3} = 0$

Solve $x^{3} = 0$ for $x$ .

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

$x = \sqrt[3]{0}$

Simplify $\sqrt[3]{0}$ .

Rewrite $0$ as $0^{3}$ .

$x = \sqrt[3]{0^{3}}$

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

$x = 0$

Set $- 3 x - 10$ equal to $0$ and solve for $x$ .

Set $- 3 x - 10$ equal to $0$ .

$- 3 x - 10 = 0$

Solve $- 3 x - 10 = 0$ for $x$ .

Add $10$ to both sides of the equation.

$- 3 x = 10$

Divide each term in $- 3 x = 10$ by $- 3$ and simplify.

Divide each term in $- 3 x = 10$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{10}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{10}{- 3}$

Divide $x$ by $1$ .

$x = \frac{10}{- 3}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{10}{3}$

Set $x^{2} - 5$ equal to $0$ and solve for $x$ .

Set $x^{2} - 5$ equal to $0$ .

$x^{2} - 5 = 0$

Solve $x^{2} - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$x^{2} = 5$

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

$x = \pm \sqrt{5}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt{5}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt{5}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt{5}, - \sqrt{5}$

The final solution is all the values that make $x^{3} (- 3 x - 10) (x^{2} - 5) = 0$ true.

$x = 0, - \frac{10}{3}, \sqrt{5}, - \sqrt{5}$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- \frac{10}{3}, 0), (\sqrt{5}, 0), (- \sqrt{5}, 0)$

Find the y-intercepts.

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

$y = - 3 {(0)}^{6} - 10 {(0)}^{5} + 15 {(0)}^{4} + 50 {(0)}^{3}$

Solve the equation.

Remove parentheses.

$y = - 3 \cdot 0^{6} - 10 {(0)}^{5} + 15 {(0)}^{4} + 50 {(0)}^{3}$

Remove parentheses.

$y = - 3 \cdot 0^{6} - 10 \cdot 0^{5} + 15 {(0)}^{4} + 50 {(0)}^{3}$

Remove parentheses.

$y = - 3 \cdot 0^{6} - 10 \cdot 0^{5} + 15 \cdot 0^{4} + 50 {(0)}^{3}$

Remove parentheses.

$y = - 3 \cdot 0^{6} - 10 \cdot 0^{5} + 15 \cdot 0^{4} + 50 \cdot 0^{3}$

Remove parentheses.

$y = - 3 {(0)}^{6} - 10 {(0)}^{5} + 15 {(0)}^{4} + 50 {(0)}^{3}$

Simplify $- 3 {(0)}^{6} - 10 {(0)}^{5} + 15 {(0)}^{4} + 50 {(0)}^{3}$ .

Simplify each term.

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

$y = - 3 \cdot 0 - 10 {(0)}^{5} + 15 {(0)}^{4} + 50 {(0)}^{3}$

Multiply $- 3$ by $0$ .

$y = 0 - 10 {(0)}^{5} + 15 {(0)}^{4} + 50 {(0)}^{3}$

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

$y = 0 - 10 \cdot 0 + 15 {(0)}^{4} + 50 {(0)}^{3}$

Multiply $- 10$ by $0$ .

$y = 0 + 0 + 15 {(0)}^{4} + 50 {(0)}^{3}$

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

$y = 0 + 0 + 15 \cdot 0 + 50 {(0)}^{3}$

Multiply $15$ by $0$ .

$y = 0 + 0 + 0 + 50 {(0)}^{3}$

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

$y = 0 + 0 + 0 + 50 \cdot 0$

Multiply $50$ by $0$ .

$y = 0 + 0 + 0 + 0$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 0 + 0$

Add $0$ and $0$ .

$y = 0 + 0$

Add $0$ and $0$ .

$y = 0$

y-intercept(s) in point form.

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

List the intersections.

x-intercept(s): $(0, 0), (- \frac{10}{3}, 0), (\sqrt{5}, 0), (- \sqrt{5}, 0)$

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

Find the x and y Intercepts f(x)=-3x^6-10x^5+15x^4+50x^3

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