Relations between Roots and Coefficients : Higher Order Equations
If α,β are ze...
Question
If α,β are zeroes of a polynomial 6x2+x–2, then the polynomial whose zeroes are 2α+3β and 3α+2β , is .
A
6x2+5x−1
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B
6x2−5x+1
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C
6x2−5x−1
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D
6x2+5x+1
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Solution
The correct option is A6x2+5x−1 y=6x2+x−2⇒α+β=−16andαβ=−26=−13Now,Letk1=2α+3βandk2=3α+2β∴k1+k2=(2α+3β+3α+2β)=5(α+β)=−56andk1×k2=(2α+3β)×(3α+2β)=6(α2+β2)+13αβ=6[(α+β)2−2αβ]+13αβ=6(α+β)2+αβ=6×136+(−13)=16−13=−16∴Requiredpolynomialwillbe:k[x2−(k1+k2)x+k1k2],wherekisnonzerorealnumber⇒k[x2+56x−16]⇒k6[6x2+5x−1]byputtingk=6,wewillgettherequiredanswer