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Hi

I don't really understand this bit:

The cubic equation x3 −3x2 + 4 = 0 has roots α , β and γ . Find the cubic equations with:


(b) roots α − 2, β − 2 and γ - 2 sigmaα = 3 sigma αβ = 0 αβγ = -4

Solution:

Sigma(α − 2) = sigma(α - 6= 3 - 6 = -3

Sigma(α − 2)(β − 2) = Sigma(αβ.) - 2sigmaα -2sigmaβ + (4x3)

= Sigma(αβ.) - 4sigmaα +12


Red parts are things I dont quite understand.


Thanks
Reply 1
Avatar for nuodai
nuodai
17
abcxyz6666
Sigma(α − 2) = sigma(α - 6= 3 - 6 = -3 Remember that ∑α=α+β+γ\sum \alpha = \alpha + \beta + \gamma, so you're summing three things -- in this case, you have α−2\alpha - 2, so you have (α−2)+(β−2)+(γ−2)(\alpha - 2) + (\beta - 2) + (\gamma - 2), which is why you multiply the 2 by 3 (i.e. it comes outside the sum).

abcxyz6666
Sigma(α − 2)(β − 2) = Sigma(αβ.) - 2sigmaα -2sigmaβ + (4x3) Same as above.

abcxyz6666
= Sigma(αβ.) - 4sigmaα +12 ∑α\newline \sum \alpha refers to sum of the roots α+β+γ\alpha + \beta + \gamma. The use of α\alpha as what to use in the sum is just convention: since β\beta is another root, we can say that ∑α=∑β=∑γ\sum \alpha = \sum \beta = \sum \gamma, and since you have -2 lots of each, that's -4*the sum.

I think the confusion here arises from the notation: it's not clear without already knowing what it means how many things you're summing. A more logical notation would be to call the roots x1...x3x_1...x_3, so that the sum of the roots is ∑i=13xi\displaystyle \sum_{i=1}^3 x_i, the product of the roots is ∑i=13xixi+1\displaystyle \sum_{i=1}^3 x_ix_{i+1} and so on.
Reply 2
Avatar for abcxyz6666
abcxyz6666
OP
2
nuodai
Remember that ∑α=α+β+γ\sum \alpha = \alpha + \beta + \gamma, so you're summing three things -- in this case, you have α−2\alpha - 2, so you have (α−2)+(β−2)+(γ−2)(\alpha - 2) + (\beta - 2) + (\gamma - 2), which is why you multiply the 2 by 3 (i.e. it comes outside the sum).

Same as above.

∑α\newline \sum \alpha refers to sum of the roots α+β+γ\alpha + \beta + \gamma. The use of α\alpha as what to use in the sum is just convention: since β\beta is another root, we can say that ∑α=∑β=∑γ\sum \alpha = \sum \beta = \sum \gamma, and since you have -2 lots of each, that's -4*the sum.

I think the confusion here arises from the notation: it's not clear without already knowing what it means how many things you're summing. A more logical notation would be to call the roots x1...x3x_1...x_3, so that the sum of the roots is ∑i=13xi\displaystyle \sum_{i=1}^3 x_i, the product of the roots is ∑i=13xixi+1\displaystyle \sum_{i=1}^3 x_ix_{i+1} and so on.




Thanks alot!!!

I understand it now.

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