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Question
If α and β are the zeros of the polynomial f(x)=3x2−6x−4, find a quadratic polynomial whose zeros are (α+β)and(α−β)
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Solution
f(x)=3x2−6x−4Roots are α and βSum =α+β=2Product =αβ=−43New Roots are (α+β) and (α−β)(α−β)2=(α+β)2−4αβ(α−β)2=4+163(α−β)=√283Sum of roots=(α+β)+(α−β)=2+√283Product of roots =(α+β)(α−β)=2√283∴f(x)=K{x2−(2+√283)x+2√283}
Roots are α and β
Sum =α+β=2
Product =αβ=−43
New Roots are (α+β) and (α−β)
(α−β)2=(α+β)2−4αβ(α−β)2=4+163(α−β)=√283
Sum of roots=(α+β)+(α−β)=2+√283
Product of roots =(α+β)(α−β)=2√283
∴f(x)=K{x2−(2+√283)x+2√283}
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