For the cubic equation x^3 - 3x^2 + 3x - 1 = 0, which of the following is true a

₹0.0

Question: For the cubic equation x^3 - 3x^2 + 3x - 1 = 0, which of the following is true about its roots?

Options:

Correct Answer: All roots are real

Solution:

The roots can be found using the Rational Root Theorem and synthetic division, confirming that all roots are real.

For the cubic equation x^3 - 3x^2 + 3x - 1 = 0, which of the following is true about its roots?

For the cubic equation x^3 - 3x^2 + 3x - 1 = 0, which of the following is true about its roots?

Step 1: Identify the cubic equation: x^3 - 3x^2 + 3x - 1 = 0.
Step 2: Use the Rational Root Theorem to find possible rational roots. The possible rational roots are factors of the constant term (-1) divided by factors of the leading coefficient (1). This gives us possible roots of ±1.
Step 3: Test the possible roots by substituting them into the equation. Start with x = 1: 1^3 - 3(1^2) + 3(1) - 1 = 1 - 3 + 3 - 1 = 0. So, x = 1 is a root.
Step 4: Use synthetic division to divide the cubic equation by (x - 1) to find the other roots. This gives us a quadratic equation: x^2 - 2x + 1 = 0.
Step 5: Factor the quadratic equation: (x - 1)(x - 1) = 0. This shows that x = 1 is a repeated root.
Step 6: Conclude that the cubic equation has one real root (x = 1) with a multiplicity of 3.

Cubic Equations – Understanding the properties and behavior of cubic equations, including the nature of their roots.
Rational Root Theorem – A method used to identify possible rational roots of a polynomial equation.
Synthetic Division – A simplified method for dividing polynomials, often used to test potential roots.
Real vs. Complex Roots – Differentiating between real and complex roots of polynomial equations.