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Understanding the degree of a polynomial is crucial for analyzing its properties and behavior. The degree is the highest exponent of the variable within the polynomial. In our example, we look at the polynomial h(x) = x^4 - x^3 + 7x^2 - 9x - 18. The highest exponent here is 4, thus making this a fourth-degree polynomial.
As a rule of thumb, the degree can tell us the number of roots the polynomial might have, as well as the potential number of turns on the graph of the polynomial. In our situation, the polynomial, being of the fourth degree, can potentially have up to four real roots and up to three turns on its graph.

The coefficients in a polynomial are the numbers that multiply the variable terms. For our example h(x), the coefficients are 1 for x^4, -1 for x^3, 7 for x^2, -9 for x, and -18 for the constant term.
Coefficients play a vital part in shaping the polynomial's graph and determining its other characteristics, such as the y-intercept. The y-intercept is always given by the constant term, which in our case is -18. So when x is zero, h(x) would be -18. Importantly, changing a coefficient can dramatically change the graph of the polynomial without affecting its degree.

A polynomial is composed of terms, which are monomials (expressions of the form ax^n, where a is a non-zero coefficient, and n is a non-negative integer—the degree of the term). The terms of the polynomial h(x) = x^4 - x^3 + 7x^2 - 9x - 18 are x^4, -x^3, 7x^2, -9x, and -18. Each term contributes to the polynomial's overall shape.
The leading term is the one with the highest degree; for h(x), it is x^4. The leading term usually dominates the behavior of the polynomial for very large or very small values of x, playing a crucial role in determining the end behavior of the polynomial's graph.

The graph of a polynomial gives a visual representation of the function over a range of values. For the given polynomial h(x), the graph would have a degree of four, which as we've discussed, means it could touch or cross the x-axis up to four times, representing up to four real zeros of the polynomial.
Furthermore, the leading coefficient (associated with the highest degree term) impacts the graph's end behavior. Since the leading coefficient of h(x) is positive, the ends of the graph point upwards as x approaches positive and negative infinity. The intermediate coefficients and signs of the terms also affect the curvature and the number of turning points on the graph. Notably, the fact that h(x)'s terms alternate in sign can suggest where the local maxima and minima may occur due to the potential for the graph to change direction at each term.