Let $A\left(x\right)$ represent a polynomial with a degree of $n-1$.
Split $A\left(x\right)$ into odd and even powers. For example:
$A\left(x\right) = 3 + 4x+6x^2+2x^3+x^4+10x^5$
$= \left(3+6x^2 + x^4\right)+x\left(4+ 2x^2 + 10x^4\right)$
More generally:
$A\left(x\right) = A_e\left(x^2\right) + \left(x\right)A_o\left(x^2\right)$
where $A_e\left(∙\right)$ are the even-numbered coefficients and $A_o\left(∙\right)$are the odd-numbered coefficients.
Are the degrees of $A_e\left(∙\right)$ and $A_o\left(∙\right)$ necessarily $≤\frac{n}{2} -1$? If so why?
This isn't homework, just a book I'm reading.