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A level Further Maths - Roots of Quadratic Equations Watch

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    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.
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    (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 \gamma when you're dealing with quadratics? They should only have \alpha, \beta since there are only 2 roots, not 3.
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    (Original post by RDKGames)
    Why do you have \gamma when you're dealing with quadratics? They should only have \alpha, \beta since there are only 2 roots, not 3.
    My bad, I meant cubics in place of quadratics!
    Thanks!
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    (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.
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    (Original post by BennClark)
    My bad, I meant cubics in place of quadratics!
    Thanks!
    That makes more sense

    Okay so write out what \displaystyle \sum \alpha \beta (\alpha + \beta) actually means

    We have \alpha \beta (\alpha + \beta) + \beta \gamma (\beta + \gamma) + \gamma \alpha (\gamma + \alpha)

    Now, as an example, if you take the first term and say \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.
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    Thanks very much guys, that makes complete sense now, I was just in one of my stupid moods again ;P
 
 
 
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