If the roots of the polynomial x^3 - 3x^2 + 3x - 1 = 0 are a, b, and c, what is

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Question: If the roots of the polynomial x^3 - 3x^2 + 3x - 1 = 0 are a, b, and c, what is the value of a + b + c?

Options:

Correct Answer: 3

Solution:

By Vieta\'s formulas, the sum of the roots a + b + c = -(-3) = 3.

If the roots of the polynomial x^3 - 3x^2 + 3x - 1 = 0 are a, b, and c, what is the value of a + b + c?

If the roots of the polynomial x^3 - 3x^2 + 3x - 1 = 0 are a, b, and c, what is the value of a + b + c?

Step 1: Identify the polynomial given in the question, which is x^3 - 3x^2 + 3x - 1.
Step 2: Recognize that the polynomial is a cubic equation of the form ax^3 + bx^2 + cx + d = 0.
Step 3: Identify the coefficients from the polynomial: a = 1, b = -3, c = 3, and d = -1.
Step 4: Recall Vieta's formulas, which relate the coefficients of the polynomial to the sums and products of its roots.
Step 5: According to Vieta's formulas, the sum of the roots (a + b + c) of the polynomial is given by -b/a.
Step 6: Substitute the values of b and a into the formula: -(-3)/1 = 3.
Step 7: Conclude that the value of a + b + c is 3.

Vieta's Formulas – Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots.