Abstract: Mathematics is a highly interconnected field, and ideas initially developed in one context can sometimes find fruitful applications in seemingly unrelated domains. A striking example of this is the recent application of tools from free probability theory to the study of the zeros of polynomials. In the first part of the talk, I will discuss one such concept: the finite free convolution of polynomials, introduced relatively recently. This notion becomes particularly appealing when applied to hypergeometric polynomials. Remarkably, these polynomials can be represented as a finite free convolution of more elementary building blocks. This representation, combined with the preservation of real zeros and interlacing properties through free convolutions, provides an effective tool for analyzing when all roots of a particular hypergeometric polynomial are real and when they exhibit monotonicity with respect to parameters. Consequently, this approach offers a fresh perspective on the zero properties of hypergeometric polynomials.Furthermore, this representation remains valid even in the asymptotic regime, allowing us to express the limit zero distribution of generalized hypergeometric polynomials as a free convolution of more "elementary" measures. This convolution can be expressed analytically by combining some integral transforms of these measures, and it turns out that in the case of hypergeometric polynomials, some of these transforms take a particularly simple form. In the second part, I will address a seemingly disconnected problem: assume we have a sequence of polynomials whose asymptotic zero distribution is known. What can be said about the zeros of their derivatives? Especially if we differentiate each polynomial several times, proportional to their degree? This simple-to-formulate question has recently attracted the attention of several researchers. Both the problem and the methods of its solution have exciting connections with free probability, random matrices, and approximation theory on the complex plane.