Question

$\sf{If \: \alpha \: and \: \beta \: be \: two \: zeros \: of \: the \: quadratic \: polynomial} \\ \sf{ax ^{2} + bx + c, then \: evaluate : }\\ \\ \sf{(i) \: \alpha^{2} + \beta^{2} }\\ \\ \sf(ii) \: \alpha^{3} + \beta^{3} \\ \\ \sf(iii) \: \frac{1}{ \alpha^{3} }$
​

Answer 1

question:

$\sf{If \: \alpha \: and \: \beta \: be \: two \: zeros \: of \: the \: quadratic \: polynomial} \\ \sf{ax ^{2} + bx + c, then \: evaluate : }\\ \\ \sf{(i) \: \alpha^{2} + \beta^{2} }\\ \\ \sf(ii) \: \alpha^{3} + \beta^{3} \\ \\ \sf(iii) \: \frac{1}{ \alpha^{3} }</p><p>$

Answer:

we are given with Quadratic polynomial , f(x) = ax² + bx + c and α & β are zeroes.

We use the relation of coefficient and zeroes.i.e.,

$\alpha+\beta=\frac{-b}{a}$

$\alpha\,\beta=\frac{c}{a}$

Consider,

$\frac{1}{\alpha^3}+\frac{1}{\beta^3}α31+β31</p><p>[tex]\implies\frac{\alpha^3+\beta^3}{(\alpha\,\beta)^3}$

$\implies\frac{( \alpha+\beta)^3+3\alpha\,\beta(\alpha+\beta)}{(\alpha\,\beta)^3}$

$\implies\frac{(\frac{-b}{a})^3+3\frac{c}{a}(\frac{-b}{a})}{(\frac{c}{a})^3}$

$\implies\frac{\frac{-b^3+3abc}{a^3}}{\frac{c^3}{a^3}}$

$\implies\frac{-b^3+3abc}{c^3}$

$\implies\frac{3abc-b^3}{c^3}$