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Polynomial functions are a critical topic in algebra and constitute one of the foundational elements of calculus. By definition, a polynomial is an expression consisting of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. An example of a polynomial of a single indeterminate x is a_nx^n + a_{n-1}x^{n-1} + ... + a_1x + a_0, where each a represents a coefficient, and n is the polynomial's degree.

In real-world applications, polynomial functions help model various phenomena. The driver fatality rate problem outlined utilizes a cubic polynomial (N(x) = ax^3 + bx^2 + cx + d), reflecting the changing fatality rates across age groups. When we input the age group number into x, the polynomial outputs the fatality rate per 100 million vehicle miles driven—essentially translating the complexities of real-world data into a comprehensible mathematical form.

Understanding age-related fatalities in driving is crucial for developing policies that could potentially save lives. The problem provided shows that as drivers age, particularly after 65, the risk of fatality per miles driven increases, likely due to physical fragility and possibly declining cognitive and motor abilities. To translate this into a usable function for analysis, the ages are grouped and coded into numbers—where x=0 represents the 50-54 year group, x=1 for 55-59, and so forth.

With these groupings, we utilize mathematical modeling to represent the trend analytically. The polynomial function, N(x), is derived from real-world data. When analyzing this model, we're not just looking at abstract numbers but seeking to understand a pattern that affects real people's lives. By doing so, age-specific interventions and driving safety programs can be tailored to better cater to each age group's needs.

Applying calculus to real-world problems allows us to understand the dynamics of change and applied rates, which is precisely what is being done in the driver fatality rate problem. In this context, calculus is used to determine the fatality rate for drivers in different age groups. Calculus application involves not just substituting values and solving, but also understanding derivatives for rate of change or finding the area under a curve for cumulative effects (integration).

For this problem, the function N(x) is predetermined, and through substitution of specific values of x, we can calculate exact fatality rates. Moreover, if we were to dig deeper beyond the scope of this exercise, derivatives of the function could show the rate at which the fatality rate is increasing or decreasing with respect to age. Such calculus applications can provide insights into the most vulnerable age groups or the points where fatality rates start increasing more rapidly, influencing road safety policies or driver assistance systems.