Find the x and y Intercepts f(x)=-4x^2(x+6)(x^2-6)(x^2+25)(x+6)

$f (x) = - 4 x^{2} (x + 6) (x^{2} - 6) (x^{2} + 25) (x + 6)$

Find the x-intercepts.

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

$0 = - 4 x^{2} (x + 6) (x^{2} - 6) (x^{2} + 25) (x + 6)$

Solve the equation.

Rewrite the equation as $- 4 x^{2} (x + 6) (x^{2} - 6) (x^{2} + 25) (x + 6) = 0$ .

$- 4 x^{2} (x + 6) (x^{2} - 6) (x^{2} + 25) (x + 6) = 0$

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

$x^{2} = 0$

$x + 6 = 0$

$x^{2} - 6 = 0$

$x^{2} + 25 = 0$

$x + 6 = 0$

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 square 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 $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

Set $x^{2} - 6$ equal to $0$ and solve for $x$ .

Set $x^{2} - 6$ equal to $0$ .

$x^{2} - 6 = 0$

Solve $x^{2} - 6 = 0$ for $x$ .

Add $6$ to both sides of the equation.

$x^{2} = 6$

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

$x = \pm \sqrt{6}$

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.

$x = \sqrt{6}$

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

$x = - \sqrt{6}$

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

$x = \sqrt{6}, - \sqrt{6}$

Set $x^{2} + 25$ equal to $0$ and solve for $x$ .

Set $x^{2} + 25$ equal to $0$ .

$x^{2} + 25 = 0$

Solve $x^{2} + 25 = 0$ for $x$ .

Subtract $25$ from both sides of the equation.

$x^{2} = - 25$

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

$x = \pm \sqrt{- 25}$

Simplify $\pm \sqrt{- 25}$ .

Rewrite $- 25$ as $- 1 (25)$ .

$x = \pm \sqrt{- 1 (25)}$

Rewrite $\sqrt{- 1 (25)}$ as $\sqrt{- 1} \cdot \sqrt{25}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{25}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \pm i \cdot \sqrt{25}$

Rewrite $25$ as $5^{2}$ .

$x = \pm i \cdot \sqrt{5^{2}}$

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

$x = \pm i \cdot 5$

Move $5$ to the left of $i$ .

$x = \pm 5 i$

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.

$x = 5 i$

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

$x = - 5 i$

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

$x = 5 i, - 5 i$

The final solution is all the values that make $- 4 x^{2} (x + 6) (x^{2} - 6) (x^{2} + 25) (x + 6) = 0$ true.

$x = 0, - 6, \sqrt{6}, - \sqrt{6}, 5 i, - 5 i$

x-intercept(s) in point form.

x-intercept(s): $(0, 0), (- 6, 0), (\sqrt{6}, 0), (- \sqrt{6}, 0)$

Find the y-intercepts.

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

$y = - 4 {(0)}^{2} ((0) + 6) ({(0)}^{2} - 6) ({(0)}^{2} + 25) ((0) + 6)$

Solve the equation.

Remove parentheses.

$y = - 4 \cdot (0^{2} ((0) + 6) ({(0)}^{2} - 6) ({(0)}^{2} + 25) ((0) + 6))$

Remove parentheses.

$y = - 4 \cdot (0^{2} (0 + 6) ({(0)}^{2} - 6) ({(0)}^{2} + 25) ((0) + 6))$

Remove parentheses.

$y = - 4 \cdot (0^{2} (0 + 6) (0^{2} - 6) ({(0)}^{2} + 25) ((0) + 6))$

Remove parentheses.

$y = - 4 \cdot (0^{2} (0 + 6) (0^{2} - 6) (0^{2} + 25) ((0) + 6))$

Remove parentheses.

$y = - 4 \cdot (0^{2} (0 + 6) (0^{2} - 6) (0^{2} + 25) (0 + 6))$

Remove parentheses.

$y = - 4 {(0)}^{2} ((0) + 6) ({(0)}^{2} - 6) ({(0)}^{2} + 25) ((0) + 6)$

Simplify $- 4 {(0)}^{2} ((0) + 6) ({(0)}^{2} - 6) ({(0)}^{2} + 25) ((0) + 6)$ .

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

$y = - 4 \cdot 0 ((0) + 6) ({(0)}^{2} - 6) ({(0)}^{2} + 25) ((0) + 6)$

Multiply $- 4$ by $0$ .

$y = 0 ((0) + 6) ({(0)}^{2} - 6) ({(0)}^{2} + 25) ((0) + 6)$

Add $0$ and $6$ .

$y = 0 \cdot 6 ({(0)}^{2} - 6) ({(0)}^{2} + 25) ((0) + 6)$

Multiply $0$ by $6$ .

$y = 0 ({(0)}^{2} - 6) ({(0)}^{2} + 25) ((0) + 6)$

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

$y = 0 (0 - 6) ({(0)}^{2} + 25) ((0) + 6)$

Subtract $6$ from $0$ .

$y = 0 \cdot - 6 ({(0)}^{2} + 25) ((0) + 6)$

Multiply $0$ by $- 6$ .

$y = 0 ({(0)}^{2} + 25) ((0) + 6)$

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

$y = 0 (0 + 25) ((0) + 6)$

Add $0$ and $25$ .

$y = 0 \cdot 25 ((0) + 6)$

Multiply $0$ by $25$ .

$y = 0 ((0) + 6)$

Add $0$ and $6$ .

$y = 0 \cdot 6$

Multiply $0$ by $6$ .

$y = 0$

y-intercept(s) in point form.

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

List the intersections.

x-intercept(s): $(0, 0), (- 6, 0), (\sqrt{6}, 0), (- \sqrt{6}, 0)$

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

Find the x and y Intercepts f(x)=-4x^2(x+6)(x^2-6)(x^2+25)(x+6)

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