To find the sum of the roots of the polynomial \( f(t) = 4t^3 - 9t^2 + 6t - 1 \), we use Vieta's formulas. For a cubic equation \( at^3 + bt^2 + ct + d = 0 \), the sum of the roots is given by \( -\frac{b}{a} \).
["# How to Find the Sum of the Roots of a Cubic Polynomial Using Vieta’s Formulas", "Pooling knowledge from algebra and polynomial theory, identifying the sum of the roots of a cubic equation can be efficient and elegant—especially when using Vieta’s formulas. For the cubic polynomial:", "[\nf(t) = 4t^3 - 9t^2 + 6t - 1\n]", "we seek a straightforward method to compute the sum of its roots without solving the entire cubic equation. This is where Vieta’s insights shine.", "## Understanding Vieta’s Formulas for Cubic Equations", "Vieta’s formulas establish powerful relationships between the coefficients of a polynomial and the sums and products of its roots. For a general cubic equation:", "[\nat^3 + bt^2 + ct + d = 0\n]", "let the roots be ( r_1 ), ( r_2 ), and ( r_3 ). Vieta’s formulas tell us:", "- Sum of roots: ( r_1 + r_2 + r_3 = -\frac{b}{a} )\n- Sum of products of roots two at a time: ( r_1r_2 + r_2r_3 + r_3r_1 = \frac{c}{a} )\n- Product of roots: ( r_1r_2r_3 = -\frac{d}{a} )", "These relationships are derived from expanding the factored form ( a(t - r_1)(t - r_2)(t - r_3) ) and comparing coefficients.", "## Applying Vieta’s Formula to Find the Sum of Roots", "Given our polynomial:", "[\nf(t) = 4t^3 - 9t^2 + 6t - 1\n]", "we identify the coefficients:\n- ( a = 4 ) (coefficient of ( t^3 ))\n- ( b = -9 ) (coefficient of ( t^2 ))", "Using Vieta’s formula for the sum of the roots:", "[\nr_1 + r_2 + r_3 = -\frac{b}{a} = -\frac{-9}{4} = \frac{9}{4}\n]", "Thus, the sum of the roots of ( f(t) = 4t^3 - 9t^2 + 6t - 1 ) is ( \frac{9}{4} ).", "## Why This Approach Matters", "This method saves time and avoids complex calculations such as factoring cubic polynomials or using numerical root-finding methods, which can be error-prone and time-consuming. For educators, students, and professionals working with polynomials, Vieta’s formulas offer a quick and reliable tool for analyzing root behavior directly from coefficients.", "## Conclusion", "Sum of roots for a cubic polynomial ( at^3 + bt^2 + ct + d ) is simply ( -\frac{b}{a} ). Applying this to ( f(t) = 4t^3 - 9t^2 + 6t - 1 ), we immediately find:", "[\n\ ext{Sum of roots} = \frac{9}{4}\n]", "Leverage Vieta’s formulas to uncover key properties of polynomials efficiently—whether for homework, exams, or real-world applications.", "---\nKeywords: sum of roots, Vieta's formulas, polynomial roots, cubic equation, algebra, mathematics, educational tool"]
