Chapter 5, Section 3
Polynomials and Polynomial Functions
An algebraic term is a number or a product of numbers and variables that are raised to a
power. Some examples include
3, 3x, 3ax 2 ,
1 5
xy
3
note that these are expressions without any addition or subtraction in them.
The number multiplier is call a coefficient of the variables.
A polynomial is an algebraic term or the sum of two or more algebraic terms whose
variables have whole number exponents.
examples of polynomials include
3x. x 2  3x  4, x 2 y  5
examples of terms that are not polynomials include
1
2x 1 , x 2 ,
5x 2
y
A polynomial that is a singleton is a monomial. examples: 3xy , 5x 2 , abc .
A polynomial that has two added terms is called a binomial: x  y, 5x  3, x 3  8 .
A polynomial with three added terms is called a trinomial: x 2  2xy  y 2 .
Right now, let’s look at polynomial functions…recall from 2.5 that a function is a
relationship between input values (domain, “x”) and output values (range, “f (x)”) in such
a way that exactly one output or range value is assigned to each number in the domain.
We’ve seen that functions pass the Vertical Line Test.
It turns out that a polynomial function is a function whose equation is defined by a
polynomial in one variable.
The degree of a polynomial function is the exponent that is largest.
Thus f (x)  5x 4  6x 3  3x 2  2x  11 is a polynomial function of degree 4.
Polynomial functions of degree 1 are called linear functions. ( f (x) = mx + b).
Polynomial function of degree 2 are called quadratic functions. f (x) = ax 2  bx  c .
Polynomial functions of degree 3 are called cubic functions.
f (x)  x 3  8 .
Those of higher degree have no special names in general use.
It, then, becomes an issue to evaluate polynomials.
Example:
Given f (x)  2x 2  3x  3 , find f(2).
Given f ( x )  x 3  8 , find f (1).
A big deal with polynomials is combining them and simplifying them. It turns out that
this is not as easy as when working strictly with numbers.
It is not possible to combine x 2  x  2 any more simply that this quadratic expression.
Any possible way of adding produces wrong answers when evaluated for natural
numbers, much less fractions.
The most you can do is “combine like terms”…that is, add up the terms that have exactly
the same variables raised to exactly the same powers.
example
2x 3  x 2  3  5x 2  11  3x 3 can be simplified
Often, there is an issue of distributing a 1 to be dealt with…this will require care.
Example
x 2  (x  5)  x  (x 3  x 2  2) can be simplified
Practice in the homework on page 321 – choose the ones with the distributions of 1,
please. There won’t be any testing over straight addition and subtraction.