Find the x and y Intercepts f(x)=-(x^2+16)(x+2)(x^2-2)

$f (x) = - (x^{2} + 16) (x + 2) (x^{2} - 2)$

Find the x-intercepts.

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

$0 = - (x^{2} + 16) (x + 2) (x^{2} - 2)$

Solve the equation.

Rewrite the equation as $- (x^{2} + 16) (x + 2) (x^{2} - 2) = 0$ .

$- (x^{2} + 16) (x + 2) (x^{2} - 2) = 0$

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

$x^{2} + 16 = 0$

$x + 2 = 0$

$x^{2} - 2 = 0$

Set $x^{2} + 16$ equal to $0$ and solve for $x$ .

Set $x^{2} + 16$ equal to $0$ .

$x^{2} + 16 = 0$

Solve $x^{2} + 16 = 0$ for $x$ .

Subtract $16$ from both sides of the equation.

$x^{2} = - 16$

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

$x = \pm \sqrt{- 16}$

Simplify $\pm \sqrt{- 16}$ .

Rewrite $- 16$ as $- 1 (16)$ .

$x = \pm \sqrt{- 1 (16)}$

Rewrite $\sqrt{- 1 (16)}$ as $\sqrt{- 1} \cdot \sqrt{16}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{16}$

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

$x = \pm i \cdot \sqrt{16}$

Rewrite $16$ as $4^{2}$ .

$x = \pm i \cdot \sqrt{4^{2}}$

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

$x = \pm i \cdot 4$

Move $4$ to the left of $i$ .

$x = \pm 4 i$

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.

$x = 4 i$

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

$x = - 4 i$

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

$x = 4 i, - 4 i$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

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

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

$x^{2} - 2 = 0$

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

Add $2$ to both sides of the equation.

$x^{2} = 2$

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

$x = \pm \sqrt{2}$

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.

$x = \sqrt{2}$

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

$x = - \sqrt{2}$

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

$x = \sqrt{2}, - \sqrt{2}$

The final solution is all the values that make $- (x^{2} + 16) (x + 2) (x^{2} - 2) = 0$ true.

$x = 4 i, - 4 i, - 2, \sqrt{2}, - \sqrt{2}$

x-intercept(s) in point form.

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

Find the y-intercepts.

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

$y = - ({(0)}^{2} + 16) ((0) + 2) ({(0)}^{2} - 2)$

Solve the equation.

Remove parentheses.

$y = - (0^{2} + 16) ((0) + 2) ({(0)}^{2} - 2)$

Remove parentheses.

$y = - (0^{2} + 16) (0 + 2) ({(0)}^{2} - 2)$

Remove parentheses.

$y = - (0^{2} + 16) (0 + 2) (0^{2} - 2)$

Remove parentheses.

$y = - ({(0)}^{2} + 16) ((0) + 2) ({(0)}^{2} - 2)$

Simplify $- ({(0)}^{2} + 16) ((0) + 2) ({(0)}^{2} - 2)$ .

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

$y = - (0 + 16) ((0) + 2) ({(0)}^{2} - 2)$

Add $0$ and $16$ .

$y = - 1 \cdot 16 ((0) + 2) ({(0)}^{2} - 2)$

Multiply $- 1$ by $16$ .

$y = - 16 ((0) + 2) ({(0)}^{2} - 2)$

Add $0$ and $2$ .

$y = - 16 \cdot 2 ({(0)}^{2} - 2)$

Multiply $- 16$ by $2$ .

$y = - 32 ({(0)}^{2} - 2)$

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

$y = - 32 (0 - 2)$

Subtract $2$ from $0$ .

$y = - 32 \cdot - 2$

Multiply $- 32$ by $- 2$ .

$y = 64$

y-intercept(s) in point form.

y-intercept(s): $(0, 64)$

List the intersections.

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

y-intercept(s): $(0, 64)$

Find the x and y Intercepts f(x)=-(x^2+16)(x+2)(x^2-2)

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