Note that each term includes the sign that precedes it. When no sign is written explicitly, that term is positive; when it includes a minus sign, it’s a negative term.
Breaking terms into coefficient and variable
Each term in an expression can be further broken up into coefficients and variables:
The coefficient of a term is the numerical part, including the minus sign when the term is negative. It's usually listed before the variable, at the beginning of the term.
The variable part of a term is everything except the coefficient, which includes all variables and their exponents.
Identifying the coefficient and variable parts of a term is useful when you want to simplify an expression by combining like terms (as I explain in Chapter 3).
Table 2-3 shows you how to break the expression
TABLE 2-3 The Expression
| Expression | Coefficient | Variable |
|---|---|---|
|
|
-1 |
|
|
|
-8 |
|
| x | 1 | x |
| 4 | 4 | None (Constant term) |
Polynomial basics
A polynomial with a single variable (x) is any expression of the following form:
This eye-glazing definition becomes a lot clearer when you see a few sample polynomials:
As you can see, each term in a polynomial is built from a variable (usually x) raised to any non-negative exponent and then multiplied by a coefficient, with terms either added or subtracted.
For clarity, polynomials are usually written in standard form, starting with the term that has the greatest exponent of x and in descending order down to the term that has the least exponent. (Note that the constant term of a polynomial technically has an exponent of 0, so it appears as the last term of a polynomial in standard form.)
Understanding the degree of a polynomial
The degree of a polynomial is determined by the exponent of its leading term — that is, the first term of the polynomial when arranged in standard form.
The leading term of a polynomial is instrumental in naming the degree of that polynomial. Table 2-4 provides examples of the four most common polynomials.
TABLE 2-4 Polynomials of Degree 1, 2, 3, and 4
| Polynomial | Degree | Name of Degree |
|---|---|---|
|
|
1 | Linear |
|
|
2 | Quadratic |
|
|
3 | Cubic |
|
|
4 | Quartic |
For more detailed information about polynomials, flip to Chapter 11.
Counting the terms of a polynomial
Polynomials are also classified by the number of terms that they have:
A monomial is a polynomial with 1 term. For example:
A binomial is a polynomial with 2 terms. For example:
A trinomial is a polynomial with
