Read: Define and Evaluate Polynomials
Learning Objectives
- Identify the degree and leading coefficient of a polynomial
- Evaluate a polynomial for given values
A polynomial function is a function consisting of sum or difference of terms in which each term is a real number, a variable, or the product of a real number and variables with an non-negative integer exponents. Non negative integers are
, ...
You may see a resemblance between expressions and polynomials, which we have been studying in this course. Polynomials are a special sub-group of mathematical expressions and equations.
The following table is intended to help you tell the difference between what is a polynomial and what is not.
IS a Polynomial | Is NOT a Polynomial | Because |
Polynomials only have variables in the numerator | ||
Polynomials only have variables in the numerator | ||
|
Roots are equivalent to rational exponents, and polynomials only have integer exponents |

, is called a binomial. A polynomial containing three terms, such as
, is called a trinomial.
We can find the degree of a polynomial by identifying the highest power of the variable that occurs in the polynomial. The term with the highest degree is called the leading term because it is usually written first. The coefficient of the leading term is called the leading coefficient. When a polynomial is written so that the powers are descending, we say that it is in standard form. It is important to note that polynomials only have integer exponents.

Given a polynomial expression, identify the degree and leading coefficient.
- Find the highest power of x to determine the degree.
- Identify the term containing the highest power of x to find the leading term.
- Identify the coefficient of the leading term.
Example
For the following polynomials, identify the degree, the leading term, and the leading coefficient.Show Answer
- The highest power of x is , so the degree is. The leading term is the term containing that degree,. The leading coefficient is the coefficient of that term,.
- The highest power of t is , so the degree is. The leading term is the term containing that degree,. The leading coefficient is the coefficient of that term,.
- The highest power of p is , so the degree is. The leading term is the term containing that degree,, The leading coefficient is the coefficient of that term,.
In the following video example, we will identify the terms, leading coefficient, and degree of a polynomial.
The table below illustrates some examples of monomials, binomials, trinomials, and other polynomials. They are all written in standard form.
Monomials | Binomials | Trinomials | Other Polynomials |
, you usually do not write the term at all (because
times anything is
, and adding
doesn’t change the value). The last binomial above could be written as a trinomial,
.
A term without a variable is called a constant term, and the degree of that term is
. For example
is the constant term in
. You would usually say that
has no constant term or that the constant term is
.
Evaluate a polynomial
You can evaluate polynomials just as you have been evaluating expressions all along. To evaluate an expression for a value of the variable, you substitute the value for the variable every time it appears. Then use the order of operations to find the resulting value for the expression.Example
Evaluate for
.
Show Solution
Substitute
for each x in the polynomial.
Following the order of operations, evaluate exponents first.
Multiply
times
, and then multiply
times
.
Change the subtraction to addition of the opposite.
Find the sum.
Answer
, for
Example
Evaluate for
.
Show Solution
Substitute
for each p in the polynomial.
Following the order of operations, evaluate exponents first and then multiply.
Add and then subtract to get
.
Answer
, for
IN the following video we show more examples of evaluating polynomials for given values of the variable.