We use Vieta's formulas to find the product of the roots. For \( h(x) = 2x^3 - 9x^2 + 12x - 4 \), the product of all roots is \( -d/a = -(-4)/2 = 2 \). Assuming the roots \( r_1, r_2, r_3 \) include a non-zero root \( r_1 \) and potentially zero roots, we check if zero is a root by substituting \( x = 0 \). Since \( h(0) = -4

["# Find the Product of the Roots Using Vieta’s Formulas: A Practical Example with ( h(x) = 2x^3 - 9x^2 + 12x - 4 )", "When solving polynomial equations, one important task is finding the product of the roots. For any cubic equation of the form:", "[\nh(x) = ax^3 + bx^2 + cx + d\n]", "Vieta’s formulas provide a powerful way to connect the coefficients of the polynomial to sums and products of its roots. Specifically, if the roots are ( r_1, r_2, r_3 ), Vieta’s formula tells us:", "[\nr_1 r_2 r_3 = -\frac{d}{a}\n]", "Let’s apply this to the polynomial:", "[\nh(x) = 2x^3 - 9x^2 + 12x - 4\n]", "Here, ( a = 2 ), ( b = -9 ), ( c = 12 ), and ( d = -4 ).", "The product of all three roots is:", "[\nr_1 r_2 r_3 = -\frac{d}{a} = -\left( \frac{-4}{2} \right) = 2\n]", "### Are Any of the Roots Zero?", "Now, suppose the polynomial might have a root at ( x = 0 ). We check this by evaluating ( h(0) ):", "[\nh(0) = 2(0)^3 - 9(0)^2 + 12(0) - 4 = -4 <br/>\ne 0\n]", "Since ( h(0) <br/>\ne 0 ), ( x = 0 ) is not a root, which confirms that none of the roots are zero. Therefore, the product calculated via Vieta’s formula — ( -\frac{d}{a} ) — is valid and equal to the actual product ( r_1 r_2 r_3 = 2 ).", "### Why This Matters", "Understanding the product of the roots helps in solving equations, verifying factorizations, and analyzing polynomial behavior. Using Vieta’s formulas eliminates the need to explicitly find the roots, saving time and effort, especially for higher-degree polynomials.", "---", "Conclusion:\nFor ( h(x) = 2x^3 - 9x^2 + 12x - 4 ), the product of the roots is confidently determined using Vieta’s formula as ( -\frac{-4}{2} = 2 ), confirmed by checking that zero is not a root. This method remains a cornerstone of algebraic efficiency in polynomial analysis.", "---", "Keywords: Vieta’s formulas, product of roots, cubic polynomial, ( h(x) = 2x^3 - 9x^2 + 12x - 4 ), root product, algebraic methods, polynomial roots, mathematical formulas."]