Polynomials
A term is an expression containing a number or the product of a number
and one or more variables raised to powers. Some examples of terms are
4x3, -x2y3, 6ab, and -2.
A polynomial is a single term or a finite sum of terms. The powers of
the variables in a polynomial must be positive integers. For example,
4x3 + (-15x2) + x + (-2)
is a polynomial. Because it is simpler to write addition of a negative as
subtraction, this polynomial is usually written as
4x3 -15x2 + x -2
The degree of a polynomial in one variable is the highest power of the
variable in the polynomial. So 4x3 -15x2 + x -2 has degree
3 and 7w -w2 has degree 2.
The degree of a term is the power of the variable in the term. Because
the last term has no variable, its degree is 0.
| 4x3 |
-15x2 |
+ x |
-2 |
| ↓ |
↓ |
↓ |
↓ |
| Third-degree term |
Second-degree term |
First-degree term |
Zero-degree term |
A single number is called a constant and so the last term is the constant
term. The degree of a polynomial consisting of a single number such as 8 is 0.
The number preceding the variable in each term is called the coefficient of a
variable or the coefficient of that term. In 4x3 -15x2 + x
-2 the coefficient of x3 is 4, the coefficient of x2 is
-15, and the coefficient of x is 1 because x = 1 · x.
Identifying coefficients
Determine the coefficients of x3 and x2 in each
polynomial:
a) x3 + 5x2 - 6
b) 4x6 - x3 + x
Solution
a) Write the polynomial as 1 · x3 + 5x2 - 6 to see that
the coefficient of x3 is 1 and the coefficient of x2 is 5.
b) The x2-term is missing in 4x6 - x3 + x.
Because 4x6 - x3 + x can be written as
4x6 - 1 · x3 + 0 · 4x6 - x2 + x,
the coefficient of x3 is -1 and the coefficient of x2
is 0.
For simplicity we generally write polynomials with the exponents decreasing
from left to right and the constant term last. So we write
x3 - 4x2 + 5x + 1 rather than - 4x2 +
1 + 5x + x3
When a polynomial is written with decreasing exponents, the coefficient of
the first term is called leading coefficient.
Certain polynomials are given special names. A monomial is a
polynomial that has one term, a binomial is a polynomial that has two
terms, and a trinomial is a polynomial that has three terms. For example,
3x5 is a monomial, 2x - 1 is a binomial, and 4x6 -3x + 2
is a trinomial.
Types of polynomials
Identify each polynomial as a monomial, binomial, or trinomial and state its
degree.
a) 5x2 - 7x3 + 2
b) x43 - x2
c) 5x
d) -12
Solution
a) The polynomial 5x2 - 7x3 + 2 is a third-degree
trinomial.
b) The polynomial x43 - x2 is a binomial with degree
43.
c) Because 5x = 5x1, this polynomial is a monomial with degree 1.
c) The polynomial -12 is a monomial with degree 0.
|
