Let α,β and γ be the roots of the equation $$x3$$−$$x2+3x+1=0$$ . Then the value of γ β α$$+2$$β$$+2$$γα γ β$$+2$$α$$+2$$γβ α γ$$+2$$α$$+2$$β is equal to

Question

Let α,β and γ be the roots of the equation $$x3$$−$$x2+3x+1=0$$ . Then the value of γ    β    α$$+2$$β$$+2$$γα    γ    β$$+2$$α$$+2$$γβ    α    γ$$+2$$α$$+2$$β is equal to

Infinity Learn · Accepted Answer

Δ$$=$$γβα$$+2$$β$$+2$$γαγβ$$+2$$α$$+2$$γβαγ$$+2$$α$$+2$$β Applying $$C3$$→$$C3$$−$$2C1$$−$$2C2$$, we get Δ$$=$$γβααγββαγ$$=($$α$$+$$β$$+$$γ$$)$$α$$2+$$β$$2+$$γ$$2$$−αβ−βγ−γα$$=($$α$$+$$β$$+$$γ$$)($$α$$+$$β$$+$$γ$$)2$$−$$3($$αβ$$+$$βγ$$+$$γα$$)=1$$×$$12$$−$$3$$×$$(3)=$$−$$8$$

$Let α, β and γ be the roots of the equation x^{3} - x^{2} + 3 x + 1 = 0 . Then the value of |\begin{array}{lll} γ β α + 2 β + 2 γ \\ α γ β + 2 α + 2 γ \\ β α γ + 2 α + 2 β \end{array}| is equal to$

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Let α,β and γ be the roots of the equation x3−x2+3x+1=0 . Then the value of γ β α+2β+2γα γ β+2α+2γβ α γ+2α+2β is equal to

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$Let α, β and γ be the roots of the equation x^{3} - x^{2} + 3 x + 1 = 0 . Then the value of |\begin{array}{lll} γ β α + 2 β + 2 γ \\ α γ β + 2 α + 2 γ \\ β α γ + 2 α + 2 β \end{array}| is equal to$