Polynomials: Symmetric Relations of Quadratic Roots

For $p(x) = 2x^2 - 5x + 7$, comparing with $ax^2 + bx + c$: $a = 2, b = -5, c = 7$. - Sum of zeroes: $\alpha + \beta = -\frac{b}{a} = -\frac{-5}{2} = \frac{5}{2}$ - Product of zeroes: $\alpha \beta = \frac{c}{a} = \frac{7}{2}$

$\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha} = \frac{\alpha^3 + \beta^3}{\alpha \beta}$ Using the identity $\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha \beta (\alpha + \beta)$: $\alpha^3 + \beta^3 = \left(\frac{5}{2}\right)^3 - 3\left(\frac{7}{2}\right)\left(\frac{5}{2}\right) = \frac{125}{8} - \frac{105}{4}$ $= \frac{125 - 210}{8} = -\frac{85}{8}$

$\frac{\alpha^2}{\beta} + \frac{\beta^2}{\alpha} = \frac{-85/8}{7/2} = -\frac{85}{8} \times \frac{2}{7} = -\frac{85}{28}$.