If $\alpha$ and $\beta$ are zeroes of the polynomial $5 x^{2}+3 x-7$, the value of $\frac{1}{\alpha}+\frac{1}{\beta}$ is
Explanation
Given that $\\alpha$ and $\\beta$ are zeroes of the polynomial $5 x^{2}+3 x-7$, we can write the polynomial as $5 x^{2}+3 x-7 = 5(x - \\alpha)(x - \\beta)$. Expanding the right-hand side, we get $5 x^{2}+3 x-7 = 5 x^{2} - 5 \\alpha x - 5 \\beta x + 5 \\alpha \\beta$. Equating the coefficients of the two expressions, we get $-5 \\alpha - 5 \\beta = 3$ and $5 \\alpha \\beta = -7$. We are asked to find the value of $\\frac{1}{\\alpha}+\\frac{1}{\\beta}$. We can rewrite this expression as $\\frac{\\alpha + \\beta}{\\alpha \\beta}$. We know that $\\alpha + \\beta = -\\frac{3}{5}$ and $\\alpha \\beta = -\\frac{7}{5}$. Therefore, $\\frac{1}{\\alpha}+\\frac{1}{\\beta} = \\frac{\\alpha + \\beta}{\\alpha \\beta} = \\frac{-3/5}{-7/5} = \\frac{3}{7}$
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