Which method to find the sum of solutions to a quintic is valid?

• A. Factoring
• B. Graphing
• C. Using Vieta's formulas
• D. Completing the square

Answer: (C)

Vieta's formulas connect the roots of a polynomial to its coefficients. For a quintic written as a x^5 + b x^4 + c x^3 + d x^2 + e x + f = 0, the sum of all five roots (including complex ones and repetitions) is -b/a. So you can get the exact total of the solutions directly from the leading two coefficients, without solving for the roots themselves. This approach works for any quintic, regardless of whether the polynomial factors nicely.

Factoring would only yield the roots if the polynomial actually factors, which isn’t guaranteed for quintics. Graphing might give an approximate sense of where the roots lie and thus an approximate sum, but it isn’t exact. Completing the square is a technique tied to quadratics and doesn’t extend to quintics.