Polynomials : sum of 2 roots = 0 (1 Viewer)

[jxballistic]

Hi all,

I'm having trouble with this polynomial question:


After I write out all the root-coefficient relationships I'm stuck.
Can someone please show me how to solve the question using only root-coefficient relationships (without doing long division or "inspection")

Thanks

 

[RealiseNothing]

Since two of it's roots add to 0, the odd powers of x in the polynomial must be equal. Hence:







Now x=0 is not a root, so it must be

So the two roots are:



Now you have the value of two roots, you should be able to use products/sums of roots to find the other two by solving simultaneously. Then you will have your factorisation.

 

[jxballistic]

Since two of it's roots add to 0, the odd powers of x in the polynomial must be equal. Hence:







Now x=0 is not a root, so it must be

So the two roots are:



Now you have the value of two roots, you should be able to use products/sums of roots to find the other two by solving simultaneously. Then you will have your factorisation.

Could you explain your first sentence? Sorry I don't understand

 

[RealiseNothing]

Could you explain your first sentence? Sorry I don't understand

Since we know the roots are

It follows that the sum of the odd powers is equal to 0, so in this question:



Hence:



This is why it is so:

Let A = sum of even powers and B = sum of odd powers. Then for roots

A + B = 0

A - B = 0

So B = 0, and thus the sum of the odd powers is 0.

 

[Sy123]

I was able to get it using the sum and products of roots rules. I will use roots a, b, c and d because writing alpha in latex is really annoying especially when you have to write ALOT of them. Well here we go.