Question : If α and β are the zeros of the quadratic polynomial f(x)=3x2-5x-2, then evaluate
i) α2+β2
ii)α3+β3
iii)α/β+β/α.
Doubt by Yathartha
Solution :
f(x) = 3x2-5x-2
a = 3
b = -5
c = -2
α+β = -b/a
α+β = -(-5)/3
α+β = 5/3
αβ = c/a
αβ = -2/3
i) α2+β2 = (α + β)2 -2αβ
α2+β2 = (5/3)2 -2(-2/3)
α2+β2 = 25/9 + 4/3
α2+β2 = (25+12)/9
α2+β2 = 37/9
ii) α3+β3 = (α+β)3-3αβ(α+β)
α3+β3 = (5/3)3-3(-2/3)(5/3)
α3+β3 = (125/27)+(30/9)
α3+β3 = (125+90)/27
α3+β3 = (125+90)/27
α3+β3 = 215/27
iii) α/β+β/α
= (α2+β2)/αβ
= (37/9)/(-2/3)
= (37/9)×(-3/2)
= -37/6