For the polynomial x^3 - 3x^2 + 3x - 1, which of the following is true about its

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Question: For the polynomial x^3 - 3x^2 + 3x - 1, which of the following is true about its roots?

Options:

Correct Answer: All roots are real

Solution:

The polynomial can be factored as (x - 1)^3, indicating that all roots are real and equal.

For the polynomial x^3 - 3x^2 + 3x - 1, which of the following is true about its roots?

For the polynomial x^3 - 3x^2 + 3x - 1, which of the following is true about its roots?

Step 1: Identify the polynomial given, which is x^3 - 3x^2 + 3x - 1.
Step 2: Look for a way to factor the polynomial. We can try to find a pattern or use synthetic division.
Step 3: Notice that the polynomial can be rewritten as (x - 1)(x^2 - 2x + 1).
Step 4: Recognize that x^2 - 2x + 1 can be factored further into (x - 1)(x - 1).
Step 5: Combine the factors: (x - 1)(x - 1)(x - 1) = (x - 1)^3.
Step 6: Conclude that the polynomial can be expressed as (x - 1)^3, which means it has a root at x = 1.
Step 7: Since the factor (x - 1) is repeated three times, this indicates that the root x = 1 is a triple root.
Step 8: Therefore, all roots of the polynomial are real and equal, specifically x = 1.

Polynomial Roots – Understanding how to find and analyze the roots of a polynomial, including their nature (real, complex, equal, etc.) and multiplicity.
Factoring Polynomials – The ability to factor polynomials to simplify the process of finding roots and understanding their properties.