Polynomials Question (1 Viewer)

[cutemouse]

Let α, β, γ be the roots of x³-x²+2x-1=0. Write down an equation with roots

i) α+β, β+γ, and α+γ
ii)α/(βγ), β/(αγ), and γ/(αβ)

Thanks

 

[clintmyster]

using sum of roots, -b/a = 1 = alpha + beta + gamma
alpha + beta = 1 - gamma

so now let y = 1 - x, therefore x = 1 - y and sub that in for the first question.

 

[cutemouse]

Thanks for that I'll that in a minute.

Could someone please help me with the other question please? I've been trying it for an hour and still couldn't get it. Also, how would I test/verify that I've the correct answer for these types of questions?

Thanks

 

[Timothy.Siu]

a/by=a^2/aby=-a^2, so u let y=-x^2

 

[youngminii]

Polynomial Transformations
Given some polynomial p(x), we:
1. Let x be any root of the original equation (eg. alpha, beta, gamma);
2. Let y be any root of the new equation;
3. Express x in terms of y;
4. Substitute this expression into the original equation;
5. Manipulate until a polynomial equation in y is obtained;
6. Replace y with x.
Taken straight of my tutor worksheet.

can you guys show working out.,.. its still about confusing for me :S

http://img18.imageshack.us/img18/4462/polynomialtransformatio.jpg

Just apply the same thing into this question.

 

[x.Exhaust.x]

LOL nice notes youngminii .

 

[waxwing]

Let α, β, γ be the roots of x³-x²+2x-1=0. Write down an equation with roots

i) α+β, β+γ, and α+γ
ii)α/(βγ), β/(αγ), and γ/(αβ)

Thanks

In case you still need some help, here are some ideas for the second case (also applicable to the first case, as someone has already hinted):
You can solve as follows:



it follows by symmetry that the roots of the new equation are the squares of the roots of the original polynomial.
This suggests using the substitution , BUT: note that substitution doesn't always work (why? because we will need to invert the substitution and rearrange it - in some cases that might not be possible).
In this case we know that p(x)=0 so we put into that equation, and rearrange to isolate , and then square it:



This is certainly the "best" way to do that particular question, but I wonder if you're aware of the other method, based on this approach: We denote the roots of the new polynomial as . Then we try to calculate the sum, product and pairwise product of the new roots (in all the following, the sigma notation indicates sum of all possibilities cycling through the roots):


where in the final step we have substituted for the sum and pairwise product of the roots using the coefficients of the polynomial.
To find the quantity will require an even more laborious calculation based on:



To summarize: substitution is preferable wherever possible (and cases where it's impossible are too nasty to be asked in textbooks or exams, generally), but it's important to recognize something like (I am not sure of the exact theorem): where the permutation does not alter the form of the transformation (you can easily see this is true in cases (i) and (ii) given), then the second method will *always* work, albeit it will take a long time!