At which​ x-intercept(s) does the graph touch the 

x​-axis 

and turn​ around?

To determine where the graph of \( f(x) = x^3 - 3x^2 - x + 3 \) touches the x-axis and turns around, we need to examine the multiplicity of the roots. A root with even multiplicity will touch the x-axis and turn around, while a root with odd multiplicity will cross the x-axis.

From the factorization we obtained:
\[ f(x) = (x - 3)(x + 1)(x - 1) \]

The roots are \( x = 3, x = -1, x = 1 \), and each root has a multiplicity of 1. Since all the multiplicities are odd, the graph crosses the x-axis at each intercept.

**Therefore, there are no x-intercepts where the graph touches the x-axis and turns around.**

This matches choice **B**: "There are no x-intercepts at which the graph touches the x-axis."

Would you like any further details or have any questions about this solution?

Here are 5 more questions you can ask:
1. What is the difference between even and odd multiplicities in polynomial roots?
2. How can you determine the multiplicity of a root from a polynomial?
3. Can you explain how to sketch the graph of a polynomial function?
4. What are some methods to factor higher-degree polynomials?
5. How does the Fundamental Theorem of Algebra relate to polynomial roots?

**Tip:** When analyzing polynomial roots, always consider the multiplicity to understand how the graph behaves at those points.

Explore how polynomial graphs touch or cross the x-axis at their turning points. Understand the concept of multiplicity of roots in polynomial functions with an example problem and solution.

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