Find the x and y Intercepts f(x)=-(x-3)^2(x^2+16)(x^2-6)

$f (x) = - {(x - 3)}^{2} (x^{2} + 16) (x^{2} - 6)$

Step 1

Find the x-intercepts.

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

$0 = - {(x - 3)}^{2} (x^{2} + 16) (x^{2} - 6)$

Solve the equation.

Rewrite the equation as $- {(x - 3)}^{2} (x^{2} + 16) (x^{2} - 6) = 0$ .

$- {(x - 3)}^{2} (x^{2} + 16) (x^{2} - 6) = 0$

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

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

$x^{2} + 16 = 0$

$x^{2} - 6 = 0$

Set ${(x - 3)}^{2}$ equal to $0$ and solve for $x$ .

Set ${(x - 3)}^{2}$ equal to $0$ .

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

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

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

$x - 3 = 0$

Add $3$ to both sides of the equation.

$x = 3$

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 $x^{2} - 6$ 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.

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

The final solution is all the values that make $- {(x - 3)}^{2} (x^{2} + 16) (x^{2} - 6) = 0$ true.

$x = 3, 4 i, - 4 i, \sqrt{6}, - \sqrt{6}$

x-intercept(s) in point form.

x-intercept(s): $(3, 0), (\sqrt{6}, 0), (- \sqrt{6}, 0)$

Step 2

Find the y-intercepts.

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

$y = - {((0) - 3)}^{2} ({(0)}^{2} + 16) ({(0)}^{2} - 6)$

Solve the equation.

Remove parentheses.

$y = - {(0 - 3)}^{2} ({(0)}^{2} + 16) ({(0)}^{2} - 6)$

Remove parentheses.

$y = - {(0 - 3)}^{2} (0^{2} + 16) ({(0)}^{2} - 6)$

Remove parentheses.

$y = - {(0 - 3)}^{2} (0^{2} + 16) (0^{2} - 6)$

Remove parentheses.

$y = - {((0) - 3)}^{2} ({(0)}^{2} + 16) ({(0)}^{2} - 6)$

Simplify $- {((0) - 3)}^{2} ({(0)}^{2} + 16) ({(0)}^{2} - 6)$ .

Subtract $3$ from $0$ .

$y = - {(- 3)}^{2} ({(0)}^{2} + 16) ({(0)}^{2} - 6)$

Raise $- 3$ to the power of $2$ .

$y = - 1 \cdot 9 ({(0)}^{2} + 16) ({(0)}^{2} - 6)$

Multiply $- 1$ by $9$ .

$y = - 9 ({(0)}^{2} + 16) ({(0)}^{2} - 6)$

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

$y = - 9 (0 + 16) ({(0)}^{2} - 6)$

Add $0$ and $16$ .

$y = - 9 \cdot 16 ({(0)}^{2} - 6)$

Multiply $- 9$ by $16$ .

$y = - 144 ({(0)}^{2} - 6)$

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

$y = - 144 (0 - 6)$

Subtract $6$ from $0$ .

$y = - 144 \cdot - 6$

Multiply $- 144$ by $- 6$ .

$y = 864$

y-intercept(s) in point form.

y-intercept(s): $(0, 864)$

Step 3

List the intersections.

x-intercept(s): $(3, 0), (\sqrt{6}, 0), (- \sqrt{6}, 0)$

y-intercept(s): $(0, 864)$

Step 4

Find the x and y Intercepts f(x)=-(x-3)^2(x^2+16)(x^2-6)

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