To determine which sets are equal, we analyze the elements of each set based on the conditions provided:

Set Definitions

1. Set A: $A = \{1, 2, 3\}$

   • Clearly listed elements: $1, 2, 3$.

2. Set B: $B = \{x; x^2 - 2x + 1 = 0\}$

   • Solve $x^2 - 2x + 1 = 0$: $(x - 1)^2 = 0 \implies x = 1$
   • Set $B = \{1\}$.

3. Set C: $C = \{x; x^3 - 6x^2 + 11x - 6 = 0\}$

   • Solve $x^3 - 6x^2 + 11x - 6 = 0$ by factoring: $x^3 - 6x^2 + 11x - 6 = (x - 1)(x - 2)(x - 3)$ Roots are $x = 1, 2, 3$.
   • Set $C = \{1, 2, 3\}$.

Comparison of Sets

• $A = \{1, 2, 3\}$
• $B = \{1\}$
• $C = \{1, 2, 3\}$

Conclusion

• Set A and Set C are equal because they both contain $\{1, 2, 3\}$.
• Set B is not equal to either because it only contains $\{1\}$.

Final Answer:

• Sets A and C are equal.

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Would you like a detailed explanation of the factoring process or additional clarifications?
Here are 5 related questions to consider:

1. How can we factorize a cubic polynomial?
2. Why does $(x - 1)^2 = 0$ result in only $x = 1$?
3. How do we verify if two sets are equal?
4. What are common methods for solving polynomial equations?
5. How does the Fundamental Theorem of Algebra help in solving such problems?

Tip: Always verify the roots of a polynomial by substituting them back into the original equation.