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Polynomials: Definitions / Evaluation (page 1 of 2)

Sections: Polynomial basics, Combining "like terms"

By now, you should be familiar with variables and exponents. You may have dealt with expressions like 3x⁴ or 6x. Polynomials are sums of these expressions. Each piece of the polynomial, each part that is being added, is called a "term". Polynomial terms have variables to whole-number exponents; there are no square roots of exponents, no fractional powers, and no variables in the denominator. Here are some examples:

Here is a typical polynomial:

Notice the exponents on the terms. The first term has exponent 2; the second term has an understood exponent 1; and the last term doesn't have any variable at all. Polynomials are usually written this way, with the terms written in "decreasing" order; that is, with the highest exponent first, the next highest next, and so forth, until you get down to the plain old number.

Any term that doesn't have a variable in it is called a "constant" term because, no matter what value you may put in for the variable x, that constant term will never change. In the picture above, no matter what x might be, 7 will always be just 7.

The first term in the polynomial, when it is written in decreasing order, is also the term with the biggest exponent, and is called the "leading term".

The exponent on a term tells you the "degree" of the term. For instance, the leading term in the above polynomial is a "second-degree term". The second term is a "first degree" term. The degree of the leading term tells you the degree of the whole polynomial; the polynomial above is a "second-degree polynomial". Here are a couple more examples:

• Give the degree of the following polynomial:  2x⁵ – 5x³ – 10x + 9

This polynomial has four terms, including a fifth-degree term, a third-degree term, a first-degree term, and a constant term.

This is a fifth-degree polynomial.

• Give the degree of the following polynomial:  7x⁴ + 6x² + x

This polynomial has three terms, including a fourth-degree term, a second-degree term, and a first-degree term. There is no constant term.

This is a fourth-degree polynomial.

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 2006-2008 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 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"
• a two-term polynomial, such as 2x + y or x² – 4, may also be called a "binomial"
• a three-term polynomial, such as 2x + y + z or x⁴ + 4x² – 4, may also be called a "trinomial"

I don't know if there are names for polynomials with a greater numbers of terms; I've never heard of any names other than what I've listed.

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.

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

Plug in –3 for x, remembering to be careful with parentheses and 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!

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