If α , β ,γ are the roots of $$x3+px2+q=0$$ where q ≠ $$0$$ and Δ$$=1/$$α $$1/$$β $$1/$$γ$$1/$$β $$1/$$γ $$1/$$α$$1/$$γ $$1/$$α $$1/$$β then ∆ equal

Question

If α , β ,γ  are the roots of $$x3+px2+q=0$$ where q ≠ $$0$$ and Δ$$=1/$$α $$1/$$β $$1/$$γ$$1/$$β $$1/$$γ $$1/$$α$$1/$$γ $$1/$$α $$1/$$β then ∆ equal

Infinity Learn · Accepted Answer

We have βγ$$+$$γα$$+$$αβ$$=0$$ . We can write ∆asΔ$$=1$$α$$3$$β$$3$$γ$$3$$βγγααβγααββγαββγγα$$=1$$α$$3$$β$$3$$γ$$3$$βγ$$+$$γα$$+$$αβγααβγα$$+$$αβ$$+$$βγαββγαβ$$+$$βγ$$+$$γαβγγα                             [using $$C1$$→ $$C1$$ $$+$$ $$C2$$ $$+$$ $$C3$$]$$=1$$α$$3$$β$$3$$γ$$30$$γααβ$$0$$αββγ$$0$$βγγα$$=0$$  [all zero property]

If $α, β, γ$ are the roots of $x^{3} + p x^{2} + q = 0$ where $q \neq 0$ and $Δ = |\begin{array}{l} 1 / α & 1 / β & 1 / γ \\ 1 / β & 1 / γ & 1 / α \\ 1 / γ & 1 / α & 1 / β \end{array}|$ then $∆$ equal

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If α , β ,γ are the roots of x3+px2+q=0 where q ≠ 0 and Δ=1/α 1/β 1/γ1/β 1/γ 1/α1/γ 1/α 1/β then ∆ equal

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If $α, β, γ$ are the roots of $x^{3} + p x^{2} + q = 0$ where $q \neq 0$ and $Δ = |\begin{array}{l} 1 / α & 1 / β & 1 / γ \\ 1 / β & 1 / γ & 1 / α \\ 1 / γ & 1 / α & 1 / β \end{array}|$ then $∆$ equal