Identify the Zeros and Their Multiplicities f(x)=2x^4-72x^2

$f (x) = 2 x^{4} - 72 x^{2}$

Step 1

Set $2 x^{4} - 72 x^{2}$ equal to $0$ .

$2 x^{4} - 72 x^{2} = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Rewrite $x^{4}$ as ${(x^{2})}^{2}$ .

$2 {(x^{2})}^{2} - 72 x^{2} = 0$

Let $u = x^{2}$ . Substitute $u$ for all occurrences of $x^{2}$ .

$2 u^{2} - 72 u = 0$

Factor $2 u$ out of $2 u^{2} - 72 u$ .

Factor $2 u$ out of $2 u^{2}$ .

$2 u (u) - 72 u = 0$

Factor $2 u$ out of $- 72 u$ .

$2 u (u) + 2 u (- 36) = 0$

Factor $2 u$ out of $2 u (u) + 2 u (- 36)$ .

$2 u (u - 36) = 0$

Replace all occurrences of $u$ with $x^{2}$ .

$2 x^{2} (x^{2} - 36) = 0$

Factor.

Rewrite $36$ as $6^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 6$ .

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

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

$x^{2} = 0$

$x + 6 = 0$

$x - 6 = 0$

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 $x - 6$ equal to $0$ and solve for $x$ .

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

$x - 6 = 0$

Add $6$ to both sides of the equation.

$x = 6$

The final solution is all the values that make $2 x^{2} (x^{2} - 36) = 0$ true. The multiplicity of a root is the number of times the root appears.

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

$x = - 6$ (Multiplicity of $1$ )

$x = 6$ (Multiplicity of $1$ )

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

$x = - 6$ (Multiplicity of $1$ )

$x = 6$ (Multiplicity of $1$ )

Step 3

Identify the Zeros and Their Multiplicities f(x)=2x^4-72x^2

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