Solution:** To find the sum of the roots of the polynomial \( P(x) = 3x^3 - 5x + 4 \), we use Vieta's formulas. For a cubic polynomial of the form \( ax^3 + bx^2 + cx + d \), the sum of the roots is given by \( -\frac{b}{a} \). Here, \( a = 3 \), \( b = 0 \), \( c = -5 \), and \( d = 4 \). Therefore, the sum of the roots is:

["How to Find the Sum of the Roots of the Polynomial ( P(x) = 3x^3 - 5x + 4 ) Using Vieta’s Formulas", "When working with polynomials, one of the most useful techniques for extracting key information about the roots without completely solving the equation is Vieta’s formulas. These formulas establish direct relationships between the coefficients of a polynomial and sums and products of its roots. Understanding how to apply Vieta’s formulas can save time and simplify calculations—especially for cubic equations.", "### Understanding the Polynomial Structure", "Consider a general cubic polynomial:", "[\nP(x) = ax^3 + bx^2 + cx + d\n]", "In our example, the polynomial is:", "[\nP(x) = 3x^3 + 0x^2 - 5x + 4\n]", "Here, ( a = 3 ), ( b = 0 ), ( c = -5 ), and ( d = 4 ).", "### Applying Vieta’s Formula for the Sum of the Roots", "For a cubic polynomial, Vieta’s formula tells us that the sum of the roots (whether real or complex) is given by:", "[\n\ ext{Sum of roots} = -\frac{b}{a}\n]", "This formula comes directly from the expansion of the factored form of the cubic polynomial:", "[\nP(x) = a(x - r_1)(x - r_2)(x - r_3)\n]\nwhere ( r_1, r_2, r_3 ) are the roots.", "Since the coefficient ( b ) corresponds to the sum of products of roots taken one at a time (with a negative sign), dividing by ( a ) gives the total algebraic sum of the roots.", "### Step-by-Step Calculation", "Substitute ( b = 0 ) and ( a = 3 ) into the formula:", "[\n\ ext{Sum of roots} = -\frac{b}{a} = -\frac{0}{3} = 0\n]", "### Conclusion", "The sum of the roots of the polynomial ( P(x) = 3x^3 - 5x + 4 ) is 0. This elegant result emerges directly from Vieta’s formulas, bypassing the need to compute individual roots or solve the cubic equation explicitly. Using these powerful formulas can dramatically improve efficiency and clarity in polynomial analysis—especially useful in algebra, engineering, and mathematical modeling.", "---", "Key Takeaways:\n- Use Vieta’s formulas to find root sums efficiently.\n- For cubic ( ax^3 + bx^2 + cx + d ), sum of roots = ( -\frac{b}{a} ).\n- This applies regardless of root types (real, complex, repeated).\n- Saves time compared to solving equations manually.", "For further exploration, learn how Vieta’s formulas also relate products and symmetric sums of roots—essential tools in polynomial theorem understanding."]