Question

If $\alpha ,\beta$are the roots of the equation $x^2-px+q=0,$then the values of $(\alpha +\beta )x-({\alpha }^2+{\beta }^2)x^2/2+({\alpha }^3+{\beta }^3)x^3/3+\dots .$is

Answer 1

Finding the value of $\mathbf{(}\mathit{\alpha }\mathbf{+}\mathit{\beta }\mathbf{)}\mathit{x}\mathbf{}\mathbf{-}\mathbf{}\mathbf{(}{\mathit{\alpha }}^{\mathbf{2}}\mathbf{+}{\mathit{\beta }}^{\mathbf{2}}\mathbf{)}{\mathit{x}}^{\mathbf{2}}\mathbf{/}\mathbf{2}\mathbf{}\mathbf{+}\mathbf{}\mathbf{(}{\mathit{\alpha }}^{\mathbf{3}}\mathbf{+}{\mathit{\beta }}^{\mathbf{3}}\mathbf{)}{\mathit{x}}^{\mathbf{3}}\mathbf{/}\mathbf{3}\mathbf{}\mathbf{}\mathbf{+}\mathbf{}\mathbf{\dots }\mathbf{.}$ :

Given that $\alpha ,\beta$are the roots of the equation $x^2-px+q=0,$

Sum of roots,$(\alpha +\beta )=p$

Product of roots, $\alpha \beta =q$

$(\alpha +\beta )x–({\alpha }^2+{\beta }^2)x^2/2+({\alpha }^3+{\beta }^3)x^3/3+\dots$

$$\begin{gathered} =[\alpha x-(1/2){\left(\alpha x\right)}^2+..]+[\beta x-(1/2){\left(\beta x\right)}^2+(1/3){\left(\beta x\right)}^3..] \\ =\log (1-\alpha x)+\log (1-\beta x) \\ =\log [1-(\alpha +\beta )x+\alpha \beta x^2] \\ =\log [1-px+q{x}^2] \end{gathered}$$

Hence, the value of $\mathbf{(}\mathit{\alpha }\mathbf{+}\mathit{\beta }\mathbf{)}\mathit{x}\mathbf{}\mathbf{-}\mathbf{}\mathbf{(}{\mathit{\alpha }}^{\mathbf{2}}\mathbf{+}{\mathit{\beta }}^{\mathbf{2}}\mathbf{)}{\mathit{x}}^{\mathbf{2}}\mathbf{/}\mathbf{2}\mathbf{}\mathbf{+}\mathbf{}\mathbf{(}{\mathit{\alpha }}^{\mathbf{3}}\mathbf{+}{\mathit{\beta }}^{\mathbf{3}}\mathbf{)}{\mathit{x}}^{\mathbf{3}}\mathbf{/}\mathbf{3}\mathbf{}\mathbf{}\mathbf{+}\mathbf{}\mathbf{\dots }\mathbf{.}$ is $\mathbf{}\mathit{l}\mathit{o}\mathit{g}\mathbf{}\mathbf{[}\mathbf{1}\mathbf{-}\mathit{p}\mathit{x}\mathbf{}\mathbf{+}\mathbf{}\mathit{q}{\mathit{x}}^{\mathbf{2}}\mathbf{]}$.