polynomials (1 Viewer)

[c0okies]

hey i looked at this question and went blank; can someone help?

Q2. Let x = alpha be a root oft the quadratic polynomial

P(x)=x^4+Ax^3+Bx^2+Ax+1 where (2+B)^2 does NOT = 4A^2

a) Show that alpha can not be 0 or 1 or -1

b) Show that x=1/alpha is a root

c) Deduce that if alpha is a multiple root, then its multiplicity is 2 and 4B=8+A^2

 

[SeDaTeD]

Spoiler

a) put 0, 1 and -1 into the polynomial and just show that it's not zero.
b) put 1/alpha into the polynomial, take a factor of 1/(alpha)^4 out and use the fact that alpha is a root to show that the thing equals zero.
c) Well you know that alpha and 1/alpha are roots. You know product of roots is 1. Thus is alpha is a multiple root, it can either have multiplicity 2 or 3, since we know 1/alpha is another root and is not equal to alpha (ie. not 1). If it has multiplicity 3, then product of roots is alpha^2 =/= 1. Thus it must have multiplicity 2, and thus 1/alpha is a double root.
Rewrite P(x) as [(x - alpha)(x - 1/alpha)]^2, expand and the result should be clear when you have A and B in terms of (alpha + 1/alpha).