10. (15 points) Provide an example of a non-zero polynomial with integer coefficients, one of whose roots is the number $\cos 18^{\circ}$.
Answer: $16 x^{4}-20 x^{2}+5$

Solution I. In an isosceles triangle with the angle at the vertex $36^{\circ}$ and the lateral side 1, the height $h$ is equal to $\cos 18^{\circ}$. Let the base be $x$, and the bisector of the angle at the base cuts off a similar triangle from this triangle with the ratio of corresponding sides $\frac{1}{x}=\frac{x}{1-x}$. From this, we find $x=\frac{\sqrt{5}-1}{2}$. By the Pythagorean theorem, $h^{2}=\frac{5+\sqrt{5}}{8},\left(8 h^{2}-5\right)^{2}=5,64 h^{4}-80 h^{2}+20=0$.

![](https://cdn.mathpix.com/cropped/2024_05_06_bc3593ebed90e6cb13c5g-3.jpg?height=753&width=480&top_left_y=1371&top_left_x=822)

Solution II. The Chebyshev polynomial $T_{5}(x)$, where $T_{n}(\cos x)=\cos n x$ satisfies the condition of the problem. (Chebyshev polynomials are constructed recursively $T_{0}=1, T_{1}=x$, $\left.T_{n+2}=2 x T_{n+1}-T_{n}\right)$