Nonnegative real polynomials as sums of two squares

Prove that a polynomial $f(x)$ remains positive for all real $x$ if and only if

where $\alpha_{0}, \alpha_{1}, \alpha_{2}, \ldots , \alpha_{n}$ are certain complex numbers.

Prove that if a polynomial $P(x)$ with real coefficients takes positive values for all $x \in \mathbb {R}$, then it can be represented as the sum of squares of two polynomials with real coefficients.

If a polynomial is real and non-negative for all real $x$ then it may be written in the form $[A(x)]^2 + [B(x)]^2$, where $A(x)$ and $B(x)$ are suitable polynomials with only real coefficients.

Every polynomial which is real and non-negative for non-negative real values of $x$, may be written in the form

where $A(x), B(x), C(x), D(x)$ denote polynomials with only real coefficients.

Every polynomial $P(x)$ of $n$th degree which can assume only real and non-negative values for $-1 \leqq x \leqq 1$ may be written in the form

and in fact in such a manner that all the four terms are at most of $n$th degree.

If $P(x)$ is of degree $2m$, there exists a representation for which $B(x) = C(x) = 0$, $A(x)$ is of degree $m$ and $D(x)$ of degree $m-1$.