Question

$\sf{\alpha \: \: a nd \: \beta \: be \: twom zeros \: of \: the \: quadratic \: polynomial} \\ \sf{ax ^{2} + bx + c, then \: evaluate : }\\ \: \sf(i) \: \alpha^{2} + \beta^{2} \\ \\ \sf(ii) \: \alpha^{3} + \beta^{3} \\ \\ \sf(iii) \: \frac{1}{ \alpha^{3} }$
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Answer 1

question:

\begin{gathered}\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 > \end{gathered}Ifαandβbetwozerosofthequadraticpolynomialax2+bx+c,thenevaluate:(i)α2+β2(ii)α3+β3(iii)α31</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}α+β=a−b

\alpha\,\beta=\frac{c}{a}αβ=ac

Consider,

\frac{1}{\alpha^3}+\frac{1}{\beta^3}α31+β31 < /p > < p > [tex]\implies\frac{\alpha^3+\beta^3}{(\alpha\,\beta)^3}α31+β31α31+β31</p><p>[tex]⟹(αβ)3α3+β3

\implies\frac{( \alpha+\beta)^3+3\alpha\,\beta(\alpha+\beta)}{(\alpha\,\beta)^3}⟹(αβ)3(α+β)3+3αβ(α+β)

\implies\frac{(\frac{-b}{a})^3+3\frac{c}{a}(\frac{-b}{a})}{(\frac{c}{a})^3}⟹(ac)3(a−b)3+3ac(a−b)

\implies\frac{\frac{-b^3+3abc}{a^3}}{\frac{c^3}{a^3}}⟹a3c3a3−b3+3abc

\implies\frac{-b^3+3abc}{c^3}⟹c3−b3+3abc

\implies\frac{3abc-b^3}{c^3}⟹c33abc−b3