For the polynomial x^3 - 3x^2 + 3x - 1, what is the nature of its roots? (2020)

₹0.0

Question: For the polynomial x^3 - 3x^2 + 3x - 1, what is the nature of its roots? (2020)

Options:

Correct Answer: All real and equal

Exam Year: 2020

Solution:

The polynomial can be factored as (x-1)^3, indicating that it has one real root with multiplicity 3, hence all roots are real and equal.

For the polynomial x^3 - 3x^2 + 3x - 1, what is the nature of its roots? (2020)

For the polynomial x^3 - 3x^2 + 3x - 1, what is the nature of its roots? (2020)

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 as (x - 1)(x - 1).
Step 5: Combine the factors: (x - 1)(x - 1)(x - 1) = (x - 1)^3.
Step 6: This shows that the polynomial has one root, which is x = 1.
Step 7: Since the factor (x - 1) is repeated 3 times, we say the root has a multiplicity of 3.
Step 8: Conclude that there is one real root (x = 1) and it is equal to itself three times.

Polynomial Factorization – Understanding how to factor polynomials to determine the nature and multiplicity of roots.
Multiplicity of Roots – Recognizing that a root with multiplicity greater than one indicates that the root is repeated.
Real vs. Complex Roots – Identifying whether roots are real or complex based on the factorization of the polynomial.