What is the sum of all zeros of f(x) = 2x³ - 6x² + x - 3?

Question

Accepted Answer

3. The correct answer is 3. By Vieta's formulas, the sum of the zeros of a polynomial aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ··· equals -aₙ₋₁/aₙ. For 2x³ - 6x² + x - 3: the sum of zeros = -(-6)/2 = 6/2 = 3. This works regardless of whether the zeros are real or complex.

The value -3 applies the formula incorrectly by taking -6/2 without the negation, or confuses the constant term with the sum of zeros. The value 1/2 may come from taking the ratio of the linear coefficient to the leading coefficient (1/2), which relates to the sum of products of zeros taken two at a time. The value -1/2 has no basis in Vieta's formulas for this polynomial.

Vieta's formulas connect the coefficients of a polynomial to symmetric functions of its zeros and appear on the CLEP Precalculus exam. For a cubic ax³ + bx² + cx + d: sum of zeros = -b/a, sum of products of zeros taken two at a time = c/a, and product of all zeros = -d/a. These formulas provide information about zeros without actually finding them, which is useful for multiple-choice questions.

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