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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, and you may have dealt with expressions like 3x4 or 6x. 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.

Here is a typical polynomial:

terms

Notice the exponents on the terms. The first term has an exponent of 2; the second term has an "understood" exponent of 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 largest exponent first, the next highest next, and so forth, until you get down to the plain old number.

 

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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" or "a term of degree two". 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.

terms

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")

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 – 4x3x + 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 2x3x2 – 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!

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Cite this article as:

Stapel, Elizabeth. "Polynomials: Definitions / Evaluation." Purplemath. Available from
    http://www.purplemath.com/modules/polydefs.htm. Accessed
 

 

 

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