To determine the minimum number of turning points for a polynomial, it is essential to understand that turning points occur where the first derivative of the polynomial changes sign. A polynomial of degree $n$ can have at most $n - 1$ turning points. However, there is no strict minimum number of turning points, as it can vary from 0 up to $n - 1$, depending on the nature of the polynomial.

Let's use the given polynomial $p(x)$ to determine the minimum number of turning points.

Step-by-Step Calculation

Step 1: Write Down the Polynomial

Given polynomial: $p(x) = (x - 1)(x + 1)x(x - 2)^2(x - 4)(x - 3)^2$

Step 2: Calculate the First Derivative

To find the turning points, we need to calculate the first derivative $p'(x)$. This process involves using the product rule multiple times due to the complexity of the polynomial.

Step 3: Determine the Critical Points

Critical points are where the first derivative $p'(x)$ is zero or undefined. These points are potential turning points.

Step 4: Analyze the Critical Points

Examine each critical point to determine if it is indeed a turning point. A turning point occurs where the first derivative changes sign.

Minimum Number of Turning Points

For the polynomial $p(x)$, let's simplify the analysis by considering the nature of the roots and their multiplicities:

• Roots and Their Multiplicities:
  • $x = -1$ (multiplicity 1)
  • $x = 0$ (multiplicity 1)
  • $x = 1$ (multiplicity 1)
  • $x = 2$ (multiplicity 2)
  • $x = 3$ (multiplicity 2)
  • $x = 4$ (multiplicity 1)

The first derivative $p'(x)$ will be zero at these points due to the factors of $p(x)$. However, not all of these points will necessarily be turning points.

Multiplicity and Turning Points

• Single Multiplicity: Roots with a single multiplicity (1, -1, 0, 4) are more likely to change the sign of the derivative, leading to turning points.
• Even Multiplicity: Roots with an even multiplicity (2, 3) are less likely to change the sign of the derivative at those points. For instance, $(x - 2)^2$ and $(x - 3)^2$ tend to "flatten out" at those roots, leading to no sign change in the derivative.

Considering these points:

• $x = 2$ and $x = 3$ do not contribute to turning points because of their even multiplicities.
• Roots $x = -1, 0, 1, 4$ are the ones where the derivative is likely to change sign.

Therefore, the polynomial $p(x)$ has a minimum of 4 turning points.

Conclusion

The minimum number of turning points for the given polynomial $p(x)$ is 4. This minimum is determined by the points where the first derivative changes sign, considering the multiplicity of the roots.

Would you like more details or have any questions?

Here are 5 questions you might find interesting:

1. How do you determine the multiplicity of roots in a polynomial?
2. What are turning points and how are they calculated for a polynomial?
3. Can you explain the end behavior of a polynomial?
4. How do you factorize complex polynomials?
5. What is the significance of the quotient and remainder in polynomial division?

Tip: When analyzing turning points, always check the multiplicity of roots. Roots with an odd multiplicity are more likely to contribute to turning points.