The given polynomial function is:

$f(x) = -2x(x + 2)(x - 2)^2$

Let's analyze the function:

1. Zeros:

   • Set $f(x) = 0$ to find the zeros.
   • The factored form gives the zeros as $x = 0$, $x = -2$, and $x = 2$.

2. Multiplicity:

   • The zero at $x = 0$ has a multiplicity of 1 (since the factor is $x$).
   • The zero at $x = -2$ has a multiplicity of 1 (since the factor is $x + 2$).
   • The zero at $x = 2$ has a multiplicity of 2 (since the factor is $(x - 2)^2$).

3. Behavior near the zeros:

   • At $x = 0$, the function crosses the x-axis (since the multiplicity is odd).
   • At $x = -2$, the function also crosses the x-axis (since the multiplicity is odd).
   • At $x = 2$, the function touches the x-axis and turns around (since the multiplicity is even).

4. Degree:

   • The degree of the polynomial is $1 + 1 + 2 = 4$, so it's a degree 4 polynomial.

5. Leading Coefficient:

   • The leading coefficient is $-2$, which affects the end behavior.

6. End Behavior:

   • Since the degree is 4 (even) and the leading coefficient is negative, the end behavior is:
     • As $x \to -\infty$, $f(x) \to -\infty$.
     • As $x \to +\infty$, $f(x) \to -\infty$.

7. Y-intercept:

   • The y-intercept occurs when $x = 0$.
   • Substituting $x = 0$ into the function: $f(0) = -2(0)(0 + 2)(0 - 2)^2 = 0$
   • The y-intercept is at $(0, 0)$.

Would you like more details or any other explanations on specific aspects of this function?

Related Questions:

1. How can you sketch the graph based on this analysis?
2. What happens to the graph at each zero based on their multiplicities?
3. How do different signs of the leading coefficient affect the end behavior?
4. Can you find the critical points by taking the derivative of this function?
5. How does the degree of a polynomial function influence its overall shape?

Tip: When analyzing the graph of a polynomial, the multiplicity of each zero provides valuable information about the behavior of the function at those points.