For the equation x^3 - 3x^2 + 3x - 1 = 0, how many real roots does it have?

For the equation x^3 - 3x^2 + 3x - 1 = 0, how many real roots does it have?

Step 1: Start with the equation x^3 - 3x^2 + 3x - 1 = 0.
Step 2: Look for a way to factor the equation. Notice that it can be rewritten as (x - 1)(x^2 - 2x + 1) = 0.
Step 3: Recognize that x^2 - 2x + 1 can be factored further as (x - 1)(x - 1).
Step 4: Combine the factors: (x - 1)(x - 1)(x - 1) = (x - 1)^3.
Step 5: Set the factored equation equal to zero: (x - 1)^3 = 0.
Step 6: Solve for x: The solution is x = 1.
Step 7: Determine the number of real roots: Since (x - 1) is repeated 3 times, it has one real root with multiplicity 3.

Polynomial Roots – Understanding how to find and interpret the roots of polynomial equations, including their multiplicities.
Factoring Polynomials – The ability to factor cubic equations and recognize patterns in polynomial expressions.
Multiplicity of Roots – Recognizing that a root can have a multiplicity greater than one and how this affects the number of distinct real roots.