The equation x^3 - 3x^2 + 3x - 1 = 0 has a root at x = 1. What is the multiplici

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Question: The equation x^3 - 3x^2 + 3x - 1 = 0 has a root at x = 1. What is the multiplicity of this root? (2023)

Options:

Correct Answer: 3

Exam Year: 2023

Solution:

The polynomial can be expressed as (x - 1)^3, indicating that the root x = 1 has a multiplicity of 3.

The equation x^3 - 3x^2 + 3x - 1 = 0 has a root at x = 1. What is the multiplicity of this root? (2023)

The equation x^3 - 3x^2 + 3x - 1 = 0 has a root at x = 1. What is the multiplicity of this root? (2023)

Step 1: Start with the given equation: x^3 - 3x^2 + 3x - 1 = 0.
Step 2: Identify that x = 1 is a root of the equation.
Step 3: To find the multiplicity, we need to factor the polynomial.
Step 4: Notice that (x - 1) is a factor of the polynomial since x = 1 is a root.
Step 5: Use polynomial long division or synthetic division to divide the polynomial by (x - 1).
Step 6: After dividing, you will find that the polynomial can be expressed as (x - 1)(x^2 - 2x + 1).
Step 7: Factor the quadratic part: x^2 - 2x + 1 = (x - 1)(x - 1).
Step 8: Now, combine the factors: (x - 1)(x - 1)(x - 1) = (x - 1)^3.
Step 9: The expression (x - 1)^3 shows that the root x = 1 appears 3 times.
Step 10: Therefore, the multiplicity of the root x = 1 is 3.

Polynomial Roots – Understanding the concept of roots and their multiplicities in polynomial equations.
Factoring Polynomials – The ability to factor polynomials to identify roots and their multiplicities.