If the roots of the cubic polynomial 2t^3-15t^2+33t-20 = 0 are in A.P. , then what is the sum of two times the first root and five times the product of the other two roots ?
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Answer:
Let the polynomial be ax3+bx2+cx+d and the zeroes be α,β,γ.
Let the polynomial be ax3+bx2+cx+d and the zeroes be α,β,γ.Sum of the polynomial = α+β+γ=12=a−b
Let the polynomial be ax3+bx2+cx+d and the zeroes be α,β,γ.Sum of the polynomial = α+β+γ=12=a−bSum of the product of its zeroes taken two at a time = αβ+βγ+αγ=1−7=ac
Let the polynomial be ax3+bx2+cx+d and the zeroes be α,β,γ.Sum of the polynomial = α+β+γ=12=a−bSum of the product of its zeroes taken two at a time = αβ+βγ+αγ=1−7=acProduct of the root = αβγ=1−14=a−d
Let the polynomial be ax3+bx2+cx+d and the zeroes be α,β,γ.Sum of the polynomial = α+β+γ=12=a−bSum of the product of its zeroes taken two at a time = αβ+βγ+αγ=1−7=acProduct of the root = αβγ=1−14=a−dIf a=1,b=−2,c=−7,d=14.
Let the polynomial be ax3+bx2+cx+d and the zeroes be α,β,γ.Sum of the polynomial = α+β+γ=12=a−bSum of the product of its zeroes taken two at a time = αβ+βγ+αγ=1−7=acProduct of the root = αβγ=1−14=a−dIf a=1,b=−2,c=−7,d=14.Hence the polynomial is x3−2x2−7x+14
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