If x3-x2-5 x-1-0 has roots - - - and x3-a x2-b x-c-0 has roots - - - then the value of a-b-c is

F(x) = x^3 x^2+5 x 1=0 has roots α, β, γ So, αβγ nbsp; = 1 nbsp;⇒αβ=1γ, βγ=1α, αγ=1β The equation whose roots are nbsp;αβ, βγ, γα is given by nbsp; f ...

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Byju's Answer

Standard XII

Mathematics

Relations between Roots and Coefficients : Higher Order Equations

If x3-x2+5 x-...

Question

If $x^{3} - x^{2} + 5 x - 1 = 0$ has roots $α, β, γ$ and $x^{3} + a x^{2} + b x + c = 0$ has roots $α β, β γ, γ α$ , then the value of $(a + b + c)$ is

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Solution

The correct option is B $- 5$
$f (x) = x^{3} - x^{2} + 5 x - 1 = 0$ has roots $α, β, γ$
So,
$α β γ = 1 \Rightarrow α β = \frac{1}{γ}, β γ = \frac{1}{α}, α γ = \frac{1}{β}$
The equation whose roots are $α β, β γ, γ α$ is given by
$f (\frac{1}{x}) = 0 \Rightarrow - x^{3} + 5 x^{2} - x + 1 = 0 \Rightarrow x^{3} - 5 x^{2} + x - 1 = 0$
Comparing with $x^{3} + a x^{2} + b x + c = 0$ , we get
$\frac{1}{1} = \frac{- 5}{a} = \frac{1}{b} = \frac{- 1}{c} \Rightarrow a = - 5, b = 1, c = - 1 ∴ a + b + c = - 5$

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If x3−x2+5x−1=0 has roots α,β,γ and x3+ax2+bx+c=0 has roots αβ,βγ,γα, then the value of (a+b+c) is

If $x^{3} - x^{2} + 5 x - 1 = 0$ has roots $α, β, γ$ and $x^{3} + a x^{2} + b x + c = 0$ has roots $α β, β γ, γ α$ , then the value of $(a + b + c)$ is