Let's define an odd polynominal be a polynominal which has odd degree, and ALL of its terms have odd exponential (except the constant), for example: $x^5+x^3+1$, or $x^7+2x^5+3x^3+4x+5$.
We all know that for every cubic polynominal $ax^3+bx^2+cx+d $, there always exists a linear transformation $t=t(x)$, such that it transforms the cubic to an odd polynominal $t^3+pt+q $ of degree 3
My question is, for every polynominal of odd degree $n$ , does there always exist a linear transformation $t=t(x)$ such that it transforms this polynominal to an odd polynominal of the same degree ?
A similar definition and question for an even polynominal: for every polynominal of even degree $n$ , does there always exist a linear transformation $t=t(x)$ such that it transforms this polynominal to an even polynominal of the same degree ?
Here is a counter example: Take the polynomial $x^{5}+x^{2}+1 $ suppose that you have $a\neq0,b $ such that $P\left(at+b\right) $ is odd, then $5a^{4}b=0$ and, $a^{2}+\binom{5}{2}a^{2}b^{3}=0 $ a contradiction.