Polynomial Roots Question (Further Maths)

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beachpanda
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Could anyone point me in the right direction of how to start this?

Solve 32z3 - 14z + 3 = 0 given that one root is twice one of the others.
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Haywood1743
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(Original post by beachpanda)
Could anyone point me in the right direction of how to start this?

Solve 32z3 - 14z + 3 = 0 given that one root is twice one of the others.
use your alpha beta gama equations but have gamma equal to 2 alpha
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beachpanda
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(Original post by Haywood1743)
use your alpha beta gama equations but have gamma equal to 2 alpha
Got it - unsure where I've gone wrong here though?
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DFranklin
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(Original post by beachpanda)
Got it - unsure where I've gone wrong here though?
Don't forget you need to divide by the coefficient of z^3. (i.e. your calculations would be valid if that coefficient was 1, but it isn't).
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beachpanda
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(Original post by DFranklin)
Don't forget you need to divide by the coefficient of z^3. (i.e. your calculations would be valid if that coefficient was 1, but it isn't).
Got it - thankyou!

(Original post by Haywood1743)
use your alpha beta gama equations but have gamma equal to 2 alpha
Thankyou as well.
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beachpanda
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My textbook also says for this separate question that the sum of alpha, beta, gamma, delta for this equation is positive 2, not negative 2.

Does that look correct? I thought it was negative 2
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ghostwalker
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(Original post by beachpanda)
My textbook also says for this separate question that the sum of alpha, beta, gamma, delta for this equation is positive 2, not negative 2.

Does that look correct? I thought it was negative 2
I presume you mean the final term; the product of all four roots.

It is +2.

If you expand (x-\alpha)(x-\beta)(x-\gamma)(x-\delta) your final term is +\alpha\beta\gamma\delta

In general:

Sum of roots works out -ve
Sum of product of 2 roots works out +ve
Sum of product of 3 roots works out -ve
Sum.....4 roots is +ve
and it carries on alternating.
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beachpanda
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(Original post by ghostwalker)
I presume you mean the final term; the product of all four roots.

It is +2.

If you expand (x-\alpha)(x-\beta)(x-\gamma)(x-\delta) your final term is +\alpha\beta\gamma\delta

In general:

Sum of roots works out -ve
Sum of product of 2 roots works out +ve
Sum of product of 3 roots works out -ve
Sum.....4 roots is +ve
and it carries on alternating.
Cool got it, thanks!
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DFranklin
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(Original post by beachpanda)
Cool got it, thanks!
On a side note, I'm going to be annoying and *suggest* you think about trying to write your alphas so they look more like \alpha and less easily confused with d. If, as here, you're dealing with quartics, where a general quartic is often written "x^4 + ax^3 + bx^2 + cx + d" I feel there's quite a real risk of confusion.

As someone who struggled a lot with handwriting at A-level I know it's annoying to get these comments, but I did find myself that it was worth making the effort.
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beachpanda
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(Original post by DFranklin)
On a side note, I'm going to be annoying and *suggest* you think about trying to write your alphas so they look more like \alpha and less easily confused with d. If, as here, you're dealing with quartics, where a general quartic is often written "x^4 + ax^3 + bx^2 + cx + d" I feel there's quite a real risk of confusion.

As someone who struggled a lot with handwriting at A-level I know it's annoying to get these comments, but I did find myself that it was worth making the effort.
That's fair - will take your advice!
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