If α and β are the roots of a quadratic polynomial 3x²-6x-1, find the values of

I) α and β

2) α²+β²

3) α-β

4) α³+β³

5) α³-β³

by chrisle

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If α and β are the roots of a quadratic polynomial 3x²-6x-1, find the values of

I) α and β

2) α²+β²

3) α-β

4) α³+β³

5) α³-β³

by chrisle

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α+β=-b/a=-(-6)/3=2

α+β=2 --------------eqn(1)

now,

αβ=c/a=-1/3 -------------eqn(2)

we know that, (α+β)^2-(α-β)^2=4*αβ

(2)^2-(α-β)^2=4*(-1/3)

4-(α-β)^2=-4/3

(α-β)^2=4+4/3

(α-β)^2=16/3

α-β = √(16/3)

α-β = +-4/√3 ----------------eqn(3)

eqn(1) + eqn(3)

2α = 2+-4/√3

α = 1+-2/√3

from eqn (1)

β=2-α

β=2-(1+-2/√3)

β=1+2/√3

when we put the value of α and β ,then we can find the value of

α^2+β^2 , α^3+β^3, and α^3-β^3