Identify the Zeros and Their Multiplicities f(x)=x^4-100x^2

$f (x) = x^{4} - 100 x^{2}$

Step 1

Set $x^{4} - 100 x^{2}$ equal to $0$ .

$x^{4} - 100 x^{2} = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

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

${(x^{2})}^{2} - 100 x^{2} = 0$

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

$u^{2} - 100 u = 0$

Factor $u$ out of $u^{2} - 100 u$ .

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

$u \cdot u - 100 u = 0$

Factor $u$ out of $- 100 u$ .

$u \cdot u + u \cdot - 100 = 0$

Factor $u$ out of $u \cdot u + u \cdot - 100$ .

$u (u - 100) = 0$

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

$x^{2} (x^{2} - 100) = 0$

Factor.

Rewrite $100$ as $10^{2}$ .

$x^{2} (x^{2} - 10^{2}) = 0$

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 $b = 10$ .

$x^{2} ((x + 10) (x - 10)) = 0$

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

$x^{2} = 0$

$x + 10 = 0$

$x - 10 = 0$

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 $x + 10$ equal to $0$ and solve for $x$ .

Set $x + 10$ equal to $0$ .

$x + 10 = 0$

Subtract $10$ from both sides of the equation.

$x = - 10$

Set $x - 10$ equal to $0$ and solve for $x$ .

Set $x - 10$ equal to $0$ .

$x - 10 = 0$

Add $10$ to both sides of the equation.

$x = 10$

The final solution is all the values that make $x^{2} (x^{2} - 100) = 0$ true. The multiplicity of a root is the number of times the root appears.

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

$x = - 10$ (Multiplicity of $1$ )

$x = 10$ (Multiplicity of $1$ )

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

$x = - 10$ (Multiplicity of $1$ )

$x = 10$ (Multiplicity of $1$ )

Step 3

Identify the Zeros and Their Multiplicities f(x)=x^4-100x^2

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