Let's break down the given polynomial function and address each part:

Given function: $f(x) = -2(x - 4)(x + 3)^2$

Part (a): Finding the real zeros and their multiplicities

1. Finding the zeros: To find the zeros of the function, we set $f(x) = 0$.

   $-2(x - 4)(x + 3)^2 = 0$

   The equation is zero if any factor is zero. This gives two cases:

   • $(x - 4) = 0$ → $x = 4$
   • $(x + 3)^2 = 0$ → $x = -3$

   Zeros:

   • $x = 4$
   • $x = -3$

2. Determining multiplicity:

   • The factor $(x - 4)$ is raised to the first power, so the multiplicity of $x = 4$ is 1.
   • The factor $(x + 3)^2$ is squared, so the multiplicity of $x = -3$ is 2.

Answer for part (a):

• Real zeros: $x = 4$ and $x = -3$
• Multiplicities:
  • The multiplicity of the larger zero $x = 4$ is 1.
  • The multiplicity of the smaller zero $x = -3$ is 2.

Part (b): Does the graph cross or touch the x-axis?

• At $x = 4$ (larger zero, multiplicity 1): Since the multiplicity is odd (1), the graph crosses the x-axis at this point.

• At $x = -3$ (smaller zero, multiplicity 2): Since the multiplicity is even (2), the graph touches the x-axis at this point and does not cross it.

Answer for part (b):

• The graph crosses the x-axis at $x = 4$.
• The graph touches the x-axis at $x = -3$.

Part (c): Maximum number of turning points

The maximum number of turning points in a polynomial function is one less than the degree of the polynomial.

The degree of the polynomial is the sum of the exponents in its factored form: $(x - 4)^1 (x + 3)^2$ The degree is $1 + 2 = 3$. Therefore, the maximum number of turning points is $3 - 1 = 2$.

Answer for part (c):

• The maximum number of turning points is 2.

Part (d): End behavior and power function

To determine the end behavior, we consider the degree and leading coefficient of the polynomial.

The leading term is obtained by expanding the function: $f(x) = -2(x - 4)(x + 3)^2$

Expanding the terms: $(x + 3)^2 = x^2 + 6x + 9$ Now multiplying: $f(x) = -2(x - 4)(x^2 + 6x + 9)$ We only need the highest degree term (the leading term): $f(x) \approx -2x(x^2) = -2x^3$ Thus, the function behaves like $-2x^3$ for large values of $|x|$.

Answer for part (d):

• The power function that the graph of $f(x)$ resembles for large values of $|x|$ is $y = -2x^3$.

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Summary Answers:

(a) The real zeros are $x = 4$ and $x = -3$.

• The multiplicity of $x = 4$ is 1.
• The multiplicity of $x = -3$ is 2.

(b) The graph crosses the x-axis at $x = 4$ and touches the x-axis at $x = -3$.
(c) The maximum number of turning points is 2.
(d) The power function is $y = -2x^3$.

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Would you like more details or have any further questions?
Here are 5 related questions you could explore:

1. How do you determine the multiplicity of a zero from a polynomial?
2. Can you explain why the graph touches the x-axis for even multiplicities?
3. How do you find the degree of a polynomial from its factored form?
4. What is the relationship between the degree of a polynomial and the number of turning points?
5. How does the leading coefficient affect the end behavior of a polynomial?

Tip: The leading term of a polynomial determines its end behavior. If the leading coefficient is negative and the degree is odd, the graph will fall to the left and rise to the right.