To determine the value of a for which one zero of the polynomial a29x2 + 13x + 6a is the reciprocal of the other, we start by letting the zeros be α and β. If α is the reciprocal of β, we have:
- α * β = 1
By Vieta’s formulas, we know:
- Sum of the roots (α + β) = -b/a = -13 / a29
- Product of the roots (α * β) = c/a = 6a / a29
From the product of the roots, we set up the equation:
- α * β = 6a / a29 = 1
This can be rearranged to find a:
- 6a = a29
- 6 = 29 (since a cannot be zero)
- a = 6 / 29
Next, we substitute this value back into the equation for the sum of the roots to ensure it holds.
Thus, the final value of a for which one zero of the polynomial is the reciprocal of the other is:
- a = 6 / 29