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Question
If α, β, and γ are the roots of the polynomial equation ax3 + bx2 + cx + d = 0, find the value of `sum alpha/(betaγ)` in terms of the coefficients
Solution
The given equation is ax3 + bx2 + cx + d = 0.
÷ a ⇒ `x^3 + "b"/alpha x^2 + "c"/alphax + "d"/alpha` = 0
Let the roots be α, β, γ
α + β + γ = `- "b"/alpha`
αβ + βγ + γα = `"c/alpha`
αβγ = `- "d"/alpha`
To find:
`sum alpha/(betaγ) = alpha/(betaγ) + beta/(γalpha) + γ/(alpha beta)`
= `(alpha^2 + beta^2 + γ^2)/(alpha beta γ)`
= `((alpha + beta + γ)^2 - 2(alphabeta + betaγ + γalpha))/(alpha beta γ)`
= `((- "b"/a"a")^2 - 2("c"/"a"))/(- "d"/"a")`
= `(("b"^2 - 2"ac"))/"a"^2 xx (- "a")/"d"`
= `(2"ac" - "b"^2)/"ad"`
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