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A level Further Maths - Roots of Quadratic Equations

Hi, for Half term homework we have been asked to find the roots of specific cubics, using α,β,γ as roots.

The questions are to find ∑α^2, ∑ α^2 β^2, and ∑αβ(α + β) of Cubics, and I thought first I'd generalise in terms of α+β+γ, αβ+γα+βγ, or αβγ

As an example:
∑α^2 is equivalent to α^2 + β^2 + γ^2, which I have generalised correctly to:
(α+β+γ)^2 - 2(αβ+γα+βγ)

My problem occurs with the third part, generalising ∑αβ(α + β).
The furthest I have got is expanding the brackets, which is not helpful at all.

Sorry if this post is unclear, the topic is hard to explain!
Thanks in advance.
(edited 6 years ago)
Original post by BennClark
Hi, for Half term homework we have been asked to find the roots of specific equations, using α,β,γ as roots.

The questions are to find ∑α^2, ∑ α^2 β^2, and ∑αβ(α + β) of quadratics, and I thought first I'd generalise in terms of α+β+γ, αβ+γα+βγ, or αβγ

As an example:
∑α^2 is equivalent to α^2 + β^2 + γ^2, which I have generalised correctly to:
(α+β+γ)^2 - 2(αβ+γα+βγ)

My problem occurs with the third part, generalising ∑αβ(α + β).
The furthest I have got is expanding the brackets, which is not helpful at all.

Sorry if this post is unclear, the topic is hard to explain!
Thanks in advance.


Why do you have γ\gamma when you're dealing with quadratics? They should only have α,β\alpha, \beta since there are only 2 roots, not 3.
Reply 2
Avatar for BennClark
BennClark
OP
7
Original post by RDKGames
Why do you have γ\gamma when you're dealing with quadratics? They should only have α,β\alpha, \beta since there are only 2 roots, not 3.


My bad, I meant cubics in place of quadratics!
Thanks!
Reply 3
Avatar for ML8020
ML8020
11
Original post by BennClark
Hi, for Half term homework we have been asked to find the roots of specific equations, using α,β,γ as roots.

The questions are to find ∑α^2, ∑ α^2 β^2, and ∑αβ(α + β) of quadratics, and I thought first I'd generalise in terms of α+β+γ, αβ+γα+βγ, or αβγ

As an example:
∑α^2 is equivalent to α^2 + β^2 + γ^2, which I have generalised correctly to:
(α+β+γ)^2 - 2(αβ+γα+βγ)

My problem occurs with the third part, generalising ∑αβ(α + β).
The furthest I have got is expanding the brackets, which is not helpful at all.

Sorry if this post is unclear, the topic is hard to explain!
Thanks in advance.

I suppose you mean roots of polynomials instead of quadratic - quadratic just means two roots.

Try multiplying (α+β+γ) by (αβ+γα+βγ), you might get what you wanted... and a bit of extra stuff as well.
(edited 6 years ago)
Original post by BennClark
My bad, I meant cubics in place of quadratics!
Thanks!

That makes more sense :smile:

Okay so write out what ∑αβ(α+β)\displaystyle \sum \alpha \beta (\alpha + \beta) actually means

We have αβ(α+β)+βγ(β+γ)+γα(γ+α)\alpha \beta (\alpha + \beta) + \beta \gamma (\beta + \gamma) + \gamma \alpha (\gamma + \alpha)

Now, as an example, if you take the first term and say αβ(α+β)=αβ(α+β+γ−γ)=αβ(α+β+γ)−αβγ\alpha \beta (\alpha + \beta) = \alpha \beta (\alpha + \beta +\gamma - \gamma)= \alpha \beta (\alpha + \beta +\gamma)-\alpha \beta \gamma, and repeat similarly for the other terms, you should get something out of that fairly quick.
Reply 5
Avatar for BennClark
BennClark
OP
7
Thanks very much guys, that makes complete sense now, I was just in one of my stupid moods again ;P

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