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Understanding the degree of a polynomial function is crucial as it provides significant insights into the graph's features. The degree refers to the highest power of the variable in the polynomial. For instance, in the polynomial function
\[P(x) = x^3 - x^2 - 2x\],

the degree is 3 because the highest power of the variable x in the equation is 3. This is not only critical for identifying the maximum number of x-intercepts, or roots, but it also affects the graph's end behavior, the potential number of turning points, and the general shape of the graph.

A core rule to remember is: a polynomial of degree n can have at most n x-intercepts and at most n-1 turning points. In the case of our cubic polynomial, we could expect up to three intercepts and therefore up to two turns within its graph. As you sketch polynomial graphs, keep this rule in mind to check your work for consistency.

X-intercepts are the points where the graph of a polynomial crosses the x-axis. These are the values of x that make the function equal to zero. Solving the equation \(P(x) = x^3 - x^2 - 2x = 0\) involves finding the roots of the polynomial.

Initially, as with our given polynomial, look for common factors. Here, the common factor x can be taken out, simplifying the process. After factoring, you are left with \(x(x - 2)(x + 1) = 0\). Applying the zero product property, it is discerned that the x-intercepts are at x = 0, x = 2, and x = -1. These are essential points when graphing the polynomial, as they are your guideposts for drawing the curve.

Moreover, knowing the x-intercepts not only helps with sketching but also has practical applications in real-world problems where these intercepts can represent the occurrence of specific events or thresholds.

The end behavior of a polynomial graph refers to how the graph behaves as x approaches positive or negative infinity. This characteristic is largely determined by the leading term of the polynomial, which consists of the term with the highest power and its coefficient.

In the function \(P(x) = x^3 - x^2 - 2x\), the leading term is \(x^3\). Since the coefficient of \(x^3\) is positive, the ends of the graph will behave in a predictable manner: as x goes to positive infinity, the graph rises to positive infinity, and as x goes to negative infinity, the graph falls to negative infinity. This information tells you that the left end of the graph will be in the third or fourth quadrant while the right end will be in the first or second quadrant.

Categorically, odd degree polynomials with positive leading coefficients will have opposite end behaviors, and the reverse for negative coefficients. Even degree polynomials, conversely, have matching end behaviors, either both up or both down depending on the sign of the leading coefficient. Always start your sketch considering the end behaviors, as they provide a framework for your entire graph.