Identify the Zeros and Their Multiplicities -8x^2(x^2-7)

$- 8 x^{2} (x^{2} - 7)$

Step 1

Set $- 8 x^{2} (x^{2} - 7)$ equal to $0$ .

$- 8 x^{2} (x^{2} - 7) = 0$

Step 2

Solve for $x$ .

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

$x^{2} = 0$

$x^{2} - 7 = 0$

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 $x^{2} - 7$ equal to $0$ and solve for $x$ .

Set $x^{2} - 7$ equal to $0$ .

$x^{2} - 7 = 0$

Solve $x^{2} - 7 = 0$ for $x$ .

Add $7$ to both sides of the equation.

$x^{2} = 7$

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

$x = \pm \sqrt{7}$

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.

$x = \sqrt{7}$

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

$x = - \sqrt{7}$

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

$x = \sqrt{7}, - \sqrt{7}$

The final solution is all the values that make $- 8 x^{2} (x^{2} - 7) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = 0$ (Multiplicity of $2$ )

$x = \sqrt{7}$ (Multiplicity of $1$ )

$x = - \sqrt{7}$ (Multiplicity of $1$ )

$x = 0$ (Multiplicity of $2$ )

$x = \sqrt{7}$ (Multiplicity of $1$ )

$x = - \sqrt{7}$ (Multiplicity of $1$ )

Step 3