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The coefficients of a polynomial play a fundamental role in defining its shape and behavior. Every term in a polynomial is made up of two parts: a coefficient and a variable raised to a power. The coefficient is a numeric factor that multiplies the variable part. For instance, in the polynomial 3x^5 - 19x^4 + 9x^3 + 71x^2 - 84x + 20, coefficients are the numbers that come before the variables, specifically 3, -19, 9, 71, -84, and the constant term 20, which can be viewed as the coefficient of x^0.

These coefficients tell us about how 'strongly' each power of the variable influences the overall function. A large coefficient on a high-degree term can dominate the behavior of the polynomial for large values of x. Conversely, smaller coefficients mean the term has a less pronounced effect. The coefficient of the highest degree term also helps to determine the end behavior as x goes to positive or negative infinity. Understanding the coefficients gives insight into the polynomial's rate of growth or decay.

The degree of a polynomial is the highest power of the variable that appears in the function and is a key indicator of the polynomial's complexity and shape. In the given polynomial 3x^5 - 19x^4 + 9x^3 + 71x^2 - 84x + 20, the highest exponent is 5, which means the degree of this polynomial is 5.

This is significant because the degree tells us several important features about the polynomial. For example, the degree gives us a clue to the maximum number of roots or solutions the polynomial equation can have, which, in this case, is up to five. It also guides us on the potential number of turning points; a polynomial of degree n can have at most n-1 turning points. Furthermore, the degree influences the end behavior; as x increases or decreases without bound, the term with the highest degree will eventually overshadow the other terms, dictating whether the polynomial's graph rises or falls and at what rate.

Breaking down a polynomial into its individual parts, we find terms, which are the building blocks of a polynomial equation. A term is a product of a coefficient and a variable raised to an exponent. Each term in a polynomial is separated by a + or - sign. The given polynomial 3x^5 - 19x^4 + 9x^3 + 71x^2 - 84x + 20 consists of six terms: 3x^5, -19x^4, 9x^3, 71x^2, -84x, and 20.

When we analyze polynomial terms, we look at three things: the coefficient (numeric factor), the variable (often x), and the exponent (degree). Different terms have different degrees, which influence where the polynomial bends or turns on its graph. Terms without variables are constants and represent the y-intercept of the polynomial's graph when plotted. By examining individual terms, students can piece together the overall behavior of the polynomial function and predict the shape of its graph.