If alpha , beta , gamma are roots of the equation x3 - x2 - 1 = 0 , then - Maths - Mathematical Induction and Binomial Theorem

Question

If alpha , beta , gamma are roots of the equation x3 -
x2 - 1 = 0 , then find the value of ( 1 +
alpha)/(1-alpha) + (1+ beta)/(1-beta) + ( 1+ gamma)/(1-gamma)

Ishwarmani · Accepted Answer

Dear
Student,
Please find
below the solution to the asked query:

As, α,β and γ are the roots of x3-x2-1=0So, α+β+γ=1,αβγ=1 and αβ+βγ+αγ=0           
iAs, α+β+γ=-ba,αβγ=-da and αβ+βγ+αγ=caNow, 1+α1-α+1+β1-β+1+γ1-γ=1+α1-β1-γ+1+β1-α1-γ+1+γ1-α1-β1-α1-β1-γ=1-β-γ+βγ+α-αβ-αγ+αβγ+1-α-γ+αγ+β-αβ-βγ+αβγ+1-α-β+αβ+γ-αγ-βγ+αβγ1-β-γ+βγ-α+αβ+αγ-αβγ=3-α+β+γ-αβ+βγ+αγ+3αβγ1-α+β+γ+αβ+βγ+αγ-αβγ=3-1-0+311-1+0-1=5-1=-5

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If alpha , beta , gamma are roots of the equation x3 - x2 - 1 = 0 , then find the value of ( 1 + alpha)/(1-alpha) + (1+ beta)/(1-beta) + ( 1+ gamma)/(1-gamma)

If alpha , beta , gamma are roots of the equation x³ - x² - 1 = 0 , then find the value of ( 1 + alpha)/(1-alpha) + (1+ beta)/(1-beta) + ( 1+ gamma)/(1-gamma)