Hi
If P(x) is monic of degree 7, if P(x) is odd, prove P(x) = ax + bx^3 + Cx^5 + x^7.
I think we need a reasoning where the eqn. has infintely many roots so it's the zero polynomial or something?
Could someone please help?
Thanx a lot!
Hi
If P(x) is monic of degree 7, if P(x) is odd, prove P(x) = ax + bx^3 + Cx^5 + x^7.
I think we need a reasoning where the eqn. has infintely many roots so it's the zero polynomial or something?
Could someone please help?
Thanx a lot!
No, it's not a zero-polynomial.
A monic 7th-degree polynomial would look like this:
.
(You should write polynomials in standard form, ie. with exponents in descending order. I'm changing the coefficients to .)
If P(x) is odd, then P(-x) = -P(x):
In order for P(-x) to equal -P(x),
,
so
.
01