Questions: Suppose that the polynomial function f is defined as follows: f(x) = x^2 + (x-1)^2 (x+1)^2 List each zero of f according to its multiplicity in the categories below. If there is more than one answer for a multiplicity, separate them with commas. If there is no answer, click on "None." Zero(s) of multiplicity one: [] Zero(s) of multiplicity two: [] Zero(s) of multiplicity three: []

Solution Steps

To find the zeros of the polynomial function \( f(x) = x^2 + (x-1)^2 (x+1)^2 \) and their multiplicities, we need to:

1. Expand the polynomial to identify the terms.
2. Factorize the polynomial to find the zeros.
3. Determine the multiplicity of each zero by examining the factors.

The polynomial function is given by: \[ f(x) = x^2 + (x-1)^2 (x+1)^2 \]

First, we expand the polynomial: \[ f(x) = x^2 + (x-1)^2 (x+1)^2 \] \[ f(x) = x^2 + (x^2 - 2x + 1)(x^2 + 2x + 1) \] \[ f(x) = x^2 + (x^4 + 2x^3 + x^2 - 2x^3 - 4x^2 - 2x + x^2 + 2x + 1) \] \[ f(x) = x^2 + x^4 - 2x^2 + 1 \] \[ f(x) = x^4 - x^2 + 1 \]

We solve for the zeros of the polynomial \( f(x) = x^4 - x^2 + 1 \). The roots are: \[ x = -\frac{\sqrt{3}}{2} + \frac{i}{2}, \quad x = \frac{\sqrt{3}}{2} - \frac{i}{2}, \quad x = -\frac{\sqrt{3}}{2} - \frac{i}{2}, \quad x = \frac{\sqrt{3}}{2} + \frac{i}{2} \]

Each of these roots has a multiplicity of 1.

Final Answer