Find all polynomials $\mathrm{P}$ with real coefficients such that for all real $x>0$ we have:

$$
\left|P(x) P\left(\frac{1}{x}\right)\right| \leqslant 1
$$

## - Complex Numbers and the Fundamental Theorem of Algebra -

Complex Numbers We have seen that there exist polynomials in $\mathbb{R}[X]$ that do not have real roots. One of the interests in introducing complex numbers (and this is the perspective from which they were introduced in the 16th century) is to overcome this difficulty through the Fundamental Theorem of Algebra (stated by d'Alembert and proved by Gauss).

Definition 3.45. Let $\mathbb{C}$ denote the set of pairs of real numbers $(a, b)$ equipped with:

1. the following addition: $(a, b) + (c, d) = (a + c, b + d)$,
2. the following multiplications: $(a, b) \times (c, d) = (ac - bd, ad + bc)$ and for $\lambda$ real, $\lambda(a, b) = (\lambda a, \lambda b)$.

We see the set of real numbers embedded in the set of complex numbers: to each real number $a$, we can associate the complex number $(a, 0)$. Finally, let us denote by $i$ the complex number $(0, 1)$. Thus, we can represent each complex number $(a, b)$ in the form $(a, b) = a(1, 0) + b(0, 1) = a + ib$.

Remark 3.46. With the multiplication rules given above, we see that $i^2 = -1$, and that $(a + bi)(c + di) = ac - bd + (ad + bc)i$. Thus, everything behaves as if $i$ were a "number" such that $i^2 = -1$ in all manipulations. In particular, $i$ is a root of the polynomial $X^2 + 1 = 0$.

Remark 3.47. Since every non-zero element of $\mathbb{C}$ has an inverse, the results from the previous sections are also valid for $\mathbb{K} = \mathbb{C}$.
Let's write $P(x)=x^{n} Q(x)$ for some integer $n \geqslant 0$ and $Q(x)$ a polynomial such that $Q(0) \neq 0$. Then $Q$ satisfies the property stated in the problem. If $Q$ is not constant, by letting $x$ tend to infinity, we see that necessarily $Q(0)=0$, which is absurd. Therefore, $Q$ is constant and $P$ is of the form $P(x)=c x^{n}$ with $c \in \mathbb{R}$ and $n \geqslant 0$. Conversely, we verify that polynomials of the form $P(x)=c x^{\mathfrak{n}}$ with $|c| \leqslant 1$ and $\mathfrak{n} \geqslant 0$ are suitable.