If α and β are the roots of the equation ax2+bx+c=0 (a≠0,a.b,c being different), then (1+α+α2)(1+β+β2) is equal to
A
Zero
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B
Positive
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C
Negative
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D
None of these
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Solution
The correct options are A Zero B Positive Given ax2+bx+c=0α,βaretheroots α+β=−ba,αβ=ca (1+α+α2)(1+β+β2) =1+α+β+α2+β2+αβ(α+β)+αβ+(αβ)2 =1+α+β+(α+β)2−2αβ+αβ(α+β)+αβ+(αβ)2 =1−ba+(−ba)2−2ca+ca(−ba)+ca+(ca)2 =1−ba+b2a2−2ca−cba2+ca+c2a2 =1−ba+b2a2−ca−cba2+c2a2 =a2−ab+b2−ac−cb+c2a2 =a2+b2+c2−ab−ac−cba2 mulitplyanddivideby2 =2a2+2b2+2c2−2ab−2ac−2cb2a2 =a2+b2−2ab+b2+c2−2bc+c2+a2−2ac2a2 =(a−b)2+(b−c)2+(c−a)22a2≥0 =positiveor0