Polynomials are favored in numerical analysis due to their straightforward structure, ease of manipulation, and well-understood behavior in various mathematical operations. Their ability to be differentiated, integrated, and approximated numerically makes them highly practical for solving problems. Additionally, polynomials are dense in large function spaces, allowing for the approximation of many functions, which is particularly useful in differential equations. This versatility is exemplified in applications like modeling the motion of a drum head using Bessel functions expressed as infinite polynomial sums. Overall, polynomials serve as a fundamental tool in numerical analysis for their flexibility and effectiveness.