Identify the Zeros and Their Multiplicities f(x)=0.5x^3+0.25x^2+0.125x+0.0625

$f (x) = 0.5 x^{3} + 0.25 x^{2} + 0.125 x + 0.0625$

Step 1

Set $0.5 x^{3} + 0.25 x^{2} + 0.125 x + 0.0625$ equal to $0$ .

$0.5 x^{3} + 0.25 x^{2} + 0.125 x + 0.0625 = 0$

Step 2

Solve for $x$ .

Factor the left side of the equation.

Factor $0.0625$ out of $0.5 x^{3} + 0.25 x^{2} + 0.125 x + 0.0625$ .

Factor $0.0625$ out of $0.5 x^{3}$ .

$0.0625 (8 x^{3}) + 0.25 x^{2} + 0.125 x + 0.0625 = 0$

Factor $0.0625$ out of $0.25 x^{2}$ .

$0.0625 (8 x^{3}) + 0.0625 (4 x^{2}) + 0.125 x + 0.0625 = 0$

Factor $0.0625$ out of $0.125 x$ .

$0.0625 (8 x^{3}) + 0.0625 (4 x^{2}) + 0.0625 (2 x) + 0.0625 = 0$

Factor $0.0625$ out of $0.0625$ .

$0.0625 (8 x^{3}) + 0.0625 (4 x^{2}) + 0.0625 (2 x) + 0.0625 (1) = 0$

Factor $0.0625$ out of $0.0625 (8 x^{3}) + 0.0625 (4 x^{2})$ .

$0.0625 (8 x^{3} + 4 x^{2}) + 0.0625 (2 x) + 0.0625 (1) = 0$

Factor $0.0625$ out of $0.0625 (8 x^{3} + 4 x^{2}) + 0.0625 (2 x)$ .

$0.0625 (8 x^{3} + 4 x^{2} + 2 x) + 0.0625 (1) = 0$

Factor $0.0625$ out of $0.0625 (8 x^{3} + 4 x^{2} + 2 x) + 0.0625 (1)$ .

$0.0625 (8 x^{3} + 4 x^{2} + 2 x + 1) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$0.0625 ((8 x^{3} + 4 x^{2}) + 2 x + 1) = 0$

Factor out the greatest common factor (GCF) from each group.

$0.0625 (4 x^{2} (2 x + 1) + 1 (2 x + 1)) = 0$

Factor.

Factor the polynomial by factoring out the greatest common factor, $2 x + 1$ .

$0.0625 ((2 x + 1) (4 x^{2} + 1)) = 0$

Remove unnecessary parentheses.

$0.0625 (2 x + 1) (4 x^{2} + 1) = 0$

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

$2 x + 1 = 0$

$4 x^{2} + 1 = 0$

Set $2 x + 1$ equal to $0$ and solve for $x$ .

Set $2 x + 1$ equal to $0$ .

$2 x + 1 = 0$

Solve $2 x + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$2 x = - 1$

Divide each term in $2 x = - 1$ by $2$ and simplify.

Divide each term in $2 x = - 1$ by $2$ .

$\frac{2 x}{2} = \frac{- 1}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{- 1}{2}$

Divide $x$ by $1$ .

$x = \frac{- 1}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$x = - \frac{1}{2}$

Set $4 x^{2} + 1$ equal to $0$ and solve for $x$ .

Set $4 x^{2} + 1$ equal to $0$ .

$4 x^{2} + 1 = 0$

Solve $4 x^{2} + 1 = 0$ for $x$ .

Subtract $1$ from both sides of the equation.

$4 x^{2} = - 1$

Divide each term in $4 x^{2} = - 1$ by $4$ and simplify.

Divide each term in $4 x^{2} = - 1$ by $4$ .

$\frac{4 x^{2}}{4} = \frac{- 1}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x^{2}}{4} = \frac{- 1}{4}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{- 1}{4}$

Simplify the right side.

Move the negative in front of the fraction.

$x^{2} = - \frac{1}{4}$

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

$x = \pm \sqrt{- \frac{1}{4}}$

Simplify $\pm \sqrt{- \frac{1}{4}}$ .

Rewrite $- \frac{1}{4}$ as ${(\frac{i}{2})}^{2}$ .

$x = \pm \sqrt{{(\frac{i}{2})}^{2}}$

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

$x = \pm \frac{i}{2}$

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.

$x = \frac{i}{2}$

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

$x = - \frac{i}{2}$

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

$x = \frac{i}{2}, - \frac{i}{2}$

The final solution is all the values that make $0.0625 (2 x + 1) (4 x^{2} + 1) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = - \frac{1}{2}$ (Multiplicity of $1$ )

$x = \frac{i}{2}$ (Multiplicity of $1$ )

$x = - \frac{i}{2}$ (Multiplicity of $1$ )

$x = - \frac{1}{2}$ (Multiplicity of $1$ )

$x = \frac{i}{2}$ (Multiplicity of $1$ )

$x = - \frac{i}{2}$ (Multiplicity of $1$ )

Step 3

Identify the Zeros and Their Multiplicities f(x)=0.5x^3+0.25x^2+0.125x+0.0625

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