4.1 polynomial

Polynomials are sums of these "variables and exponents" expressions. Each piece of the polynomial, each part that is being added, is called a "term". Polynomial terms have variables which are raised to whole-number exponents (or else the terms are just plain numbers); there are no square roots of variables, no fractional powers, and no variables in the denominator of any fractions. 

Here are some examples:

6x –2	This is NOT a polynomial term...	...because the variable has a negative exponent.
1/x2	This is NOT a polynomial term...	...because the variable is in the denominator.
sqrt(x)	This is NOT a polynomial term...	...because the variable is inside a radical.
4x2	This IS a polynomial term...	...because it obeys all the rules.

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When a term contains both a number and a variable part, the number part is called the "coefficient". The coefficient on the leading term is called the "leading" coefficient.

In the above example, the coefficient of the leading term is 4; the coefficient of the second term is 3; the constant term doesn't have a coefficient.   Copyright © Elizabeth Stapel 2000-2011 All Rights Reserved

The "poly" in "polynomial" means "many". I suppose, technically, the term "polynomial" should only refer to sums of many terms, but the term is used to refer to anything from one term to the sum of a zillion terms. However, the shorter polynomials do have their own names:

• a one-term polynomial, such as 2x or 4x², may also be called a "monomial" ("mono" meaning "one")
• a two-term polynomial, such as 2x + y or x² – 4, may also be called a "binomial" ("bi" meaning "two")
• a three-term polynomial, such as 2x + y + z or x⁴ + 4x² – 4, may also be called a "trinomial" ("tri" meaning "three")

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Polynomials are also sometimes named for their degree:

• a second-degree polynomial, such as 4x², x² – 9, or ax² + bx + c, is also called a "quadratic"
• a third-degree polynomial, such as –6x³ or x³ – 27, is also called a "cubic"
• a fourth-degree polynomial, such as x⁴ or 2x⁴ – 3x² + 9, is sometimes called a "quartic"
• a fifth-degree polynomial, such as 2x⁵ or x⁵ – 4x³ – x + 7, is sometimes called a "quintic"

There are names for some of the higher degrees, but I've never heard of any names being used other than the ones I've listed.

By the way, yes, "quad" generally refers to "four", as when an ATV is referred to as a "quad bike". For polynomials, however, the "quad" from "quadratic" is derived from the Latin for "making square". As in, if you multiply length by width (of, say, a room) to find the area in "square" units, the units will be raised to the second power. The area of a room that is 6 meters by 8 meters is 48 m². So the "quad" refers to the four corners of a square, from the geometrical origins of parabolas and early polynomials.

Evaluation 

"Evaluating" a polynomial is the same as evaluating anything else: you plug in the given value of x, and figure out what y is supposed to be. For instance:

• Evaluate 2x³ – x² – 4x + 2 at x = –3

I need to plug in "–3" for the "x", remembering to be careful with my parentheses and the negatives:
2(–3)³ – (–3)² – 4(–3) + 2   =  2(–27) – (9) + 12 + 2   =  –54 – 9 + 14   =  –63 + 14   =  –49

Always remember to be careful with the minus signs!