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A polynomial function is a mathematical expression that involves a sum of powers in one or more variables, multiplied by coefficients. For example, the general form of a polynomial function in a single variable x is given by \(f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_2 x^2 + a_1 x + a_0 \) where \(a_n, a_{n-1}, ..., a_1, a_0 \) are constants called coefficients, and n is a whole number known as the degree of the polynomial.

The behavior of polynomial functions can vary hugely depending on their degree and their coefficients. Properties such as end behavior, the number of turning points, and the number of real roots are closely linked to the degree of the polynomial. For instance, a quadratic function, which is a polynomial of degree 2, will always have a graph that is a parabola.

When graphing polynomials, the goal is to get a visual representation of the function on a coordinate system. You'll want to identify key features such as intercepts, turning points, end behavior, and any symmetries. It's helpful to start by plotting any known points, such as the y-intercept (which occurs at \(f(0)\)) and any x-intercepts or 'roots' where the polynomial equals zero.

The viewing rectangle simply determines the scale of the x-axis and y-axis. By correctly setting the viewing rectangle, in this case, [-1,6,1] by[-50,50,10], you can accurately capture the behavior of the polynomial on the graph. You should see how the graph behaves as it crosses the x-axis, indicating potential roots, and how the graph goes to infinity or negative infinity as \(x \) goes to infinity or negative infinity, respectively.

To identify the rational roots of a polynomial, the Rational Zero Theorem is extremely useful. It states that if a polynomial has rational roots, they must be of the form \( ±p/q \), where p is a factor of the constant term and q is a factor of the leading coefficient. The theorem doesn't tell you which rational numbers are roots, but it does give a list of possibilities to check.

In our exercise, applying this theorem to the polynomial \(2 x^{3} - 15 x^{2} + 22 x + 15 = 0 \) gives the list of possible rational roots such as \(±1, ±\frac{3}{2}, ±5, ±\frac{15}{2} \), and so on. To identify which of these are actual roots, these candidates would then be tested, either by synthetic division or substitution into the polynomial, or by observing where the graph crosses the x-axis.

The coefficients in a polynomial are the multipliers of the variable terms. The factors of the polynomial coefficients are significant when using the Rational Zero Theorem. As shown in the exercise, the first step in the theorem is identifying the factors of the constant term (in this case, 15) and the factors of the leading coefficient (in this case, 2).

Once you've listed these factors, they form the pool from which potential rational roots are drawn. The pool includes all possible fraction combinations of the factors of the constant term over the factors of the leading coefficient, as well as their negative counterparts. These possible roots can then be tested against the polynomial to determine if they are actual roots, further narrowing down the search for the solution of the polynomial equation.