The equation x^3 - 3x^2 + 3x - 1 = 0 has how many distinct real roots? (2022)

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Question: The equation x^3 - 3x^2 + 3x - 1 = 0 has how many distinct real roots? (2022)

Options:

Correct Answer: 1

Exam Year: 2022

Solution:

The given polynomial can be factored as (x - 1)^3 = 0, which has one distinct real root, x = 1.

The equation x^3 - 3x^2 + 3x - 1 = 0 has how many distinct real roots? (2022)

The equation x^3 - 3x^2 + 3x - 1 = 0 has how many distinct real roots? (2022)

Step 1: Start with the equation x^3 - 3x^2 + 3x - 1 = 0.
Step 2: Look for a way to factor the polynomial. Notice that it can be rewritten as (x - 1)(x^2 - 2x + 1).
Step 3: Recognize that x^2 - 2x + 1 can be factored further as (x - 1)(x - 1).
Step 4: Combine the factors to get (x - 1)(x - 1)(x - 1) or (x - 1)^3 = 0.
Step 5: Set the factored equation (x - 1)^3 = 0 to find the roots.
Step 6: Solve for x by setting x - 1 = 0, which gives x = 1.
Step 7: Determine the number of distinct real roots. Since (x - 1) is repeated three times, there is only one distinct real root, which is x = 1.

Polynomial Roots – Understanding how to find and analyze the roots of polynomial equations.
Factoring Polynomials – Recognizing how to factor polynomials to simplify the process of finding roots.
Multiplicity of Roots – Identifying the concept of root multiplicity and its impact on the number of distinct roots.