section 5.4 Learning Objectives
5.4: Introduction to Polynomials
- Determine if an expression is a polynomial
- Identify the characteristics of a polynomial
- Determine if a polynomial is a monomial, binomial, or trinomial
- Evaluate a polynomial for a specified value
- Simplify polynomials by combining like terms
- Determine the domain of a polynomial function
Polynomials are algebraic expressions that are created by combining numbers and variables using arithmetic operations such as addition, subtraction, multiplication, division, and exponentiation. You can create a polynomial by adding or subtracting terms. Polynomials are very useful in applications from science and engineering to business. You may see a resemblance between expressions, which we have been studying in this course, and polynomials. Polynomials are a special sub-group of mathematical expressions and equations.
Determine if an expression is a polynomial
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 |
| [latex]2x^2-\frac{1}{2}x -9[/latex] | [latex]\frac{2}{x^{2}}+x[/latex] | Polynomials only have variables in the numerator |
| [latex]\frac{y}{4}-y^3[/latex] | [latex]\frac{2}{y}+4[/latex] | Polynomials only have variables in the numerator |
| [latex]\sqrt{12}\left(a\right)+9[/latex] | [latex]\sqrt{a}+7[/latex] | Variables under a root are not allowed in polynomials |
Identify the characteristics of a polynomial
The basic building block of a polynomial is a monomial. A monomial is one term and can be a number, a variable, or the product of a number and variables with an exponent. The number part of the term is called the coefficient.
Examples of monomials:
- number: [latex]{2}[/latex]
- variable: [latex]{x}[/latex]
- product of number and variable: [latex]{2x}[/latex]
- product of number and variable with an exponent: [latex]{2x}^{3}[/latex]
The coefficient can be any real number, including 0. The exponent of the variable must be a whole number—0, 1, 2, 3, and so on. A monomial cannot have a variable in the denominator or a negative exponent.
For a monomial in one variable, the value of the exponent is called the degree of the monomial. Based on our exponent rules in Section 5.1, recall that [latex]x^{0}=1[/latex]. So, a monomial with no variable actually has a degree of 0. For example, we could rewrite the monomial 3 as [latex]3x^{0}[/latex].
Example 1
Identify the coefficient, variable, and degree of the variable for the following monomial terms:
1) 9
2) [latex]x[/latex]
3) [latex] \displaystyle \frac{3}{5}{{k}^{8}}[/latex]
A polynomial is a monomial or the sum or difference of two or more monomials. Each monomial is called a term of the polynomial.
The word “polynomial” has the prefix, “poly,” which means many. However, the word polynomial can be used for all numbers of terms, including only one term.
Because the exponent of the variable must be a whole number, monomials and polynomials cannot have a variable in the denominator.
Polynomials can be classified by the degree of the polynomial. The degree of a polynomial is the degree of its highest-degree term. The coefficient of the highest degree term is called the leading coefficient. So the degree of [latex]2x^{3}+3x^{2}+8x+5[/latex] is 3 and the leading coefficient is 2.
A polynomial is said to be written in standard form (or “descending order”) when the terms are arranged from the highest-degree to the lowest degree. When it is written in standard form it is easy to determine the degree of the polynomial.
If a polynomial contains a term with no variable, it is called the constant term. If the polynomial is in standard form, the constant term appears at the end. If no constant term is explicitly given, then the constant term is 0.
terminology
- Polynomial – a sum of monomials, each called a term of the polynomial
- Degree of a Polynomial – the degree of its highest-degree term
- Leading Coefficient – the coefficient of the highest-degree term
- Constant Term – the term with no variable (if no such term is written, it is 0)
Example 2
For each polynomial, determine the number of terms, the degree of the polynomial, the leading coefficient, and the constant term.
- [latex]\hspace{.05in} 7x^3-4x^2+5x+8[/latex]
- [latex]\hspace{.05in}-3x^9+2x[/latex]
- [latex]\hspace{.05in}x^{4}-x-1[/latex]
- [latex]\hspace{.05in}13-x^2+2x+\dfrac{3}{4}x^{5}+x^3[/latex]
Determine if a polynomial is a monomial, binomial, or trinomial
Some polynomials have specific names indicated by their prefix.
- Monomial—is a polynomial with exactly one term (“mono”—means one)
- Binomial—is a polynomial with exactly two terms (“bi”—means two)
- Trinomial—is a polynomial with exactly three terms (“tri”—means three)
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 |
| 15 | [latex]3y+13[/latex] | [latex]x^{3}-x^{2}+1[/latex] | [latex]5x^{4}+3x^{3}-6x^{2}+2x[/latex] |
| [latex] \displaystyle \frac{1}{2}x[/latex] | [latex]4p-7[/latex] | [latex]3x^{2}+2x-9[/latex] | [latex]\frac{1}{3}x^{5}-2x^{4}+\frac{2}{9}x^{3}-x^{2}+4x-\frac{5}{6}[/latex] |
| [latex]-4y^{3}[/latex] | [latex]3x^{2}+\frac{5}{8}x[/latex] | [latex]3y^{3}+y^{2}-2[/latex] | [latex]3t^{3}-3t^{2}-3t-3[/latex] |
| [latex]16n^{4}[/latex] | [latex]14y^{3}+3y[/latex] | [latex]a^{7}+2a^{5}-3a^{3}[/latex] | [latex]q^{7}+2q^{5}-3q^{3}+q[/latex] |
When the coefficient of a polynomial term is 0, you usually do not write the term at all (because 0 times anything is 0, and adding 0 doesn’t change the value). The last binomial above could be written as a trinomial, [latex]14y^{3}+0y^{2}+3y[/latex].
Example 3
For the following expressions, determine whether they are a polynomial. If so, categorize them as a monomial, binomial, or trinomial.
- [latex]\frac{x-3}{1-x}+x^2[/latex]
- [latex]t^2+2t-3[/latex]
- [latex]x^3+\frac{x}{8}[/latex]
- [latex]\frac{\sqrt{y}}{2}-y-1[/latex]
In the following video, you will be shown more examples of how to identify and categorize polynomials.
Evaluate a polynomial for a specified value
You can evaluate polynomials just as you have been evaluating expressions and functions all along. (Recall, we evaluated functions back in Module 3). 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 4
Given [latex]f(x)=3x^{2}-2x+1[/latex], evaluate [latex]f(-1)[/latex].
Example 5
Given [latex] f(p)= \displaystyle -\frac{2}{3}p^{4}+2p^{3}-p[/latex], evaluate [latex]f(3)[/latex].
The following video presents more examples of evaluating a polynomial for a given value.
Simplify polynomials by combining like terms
Apple and Orange
A polynomial may need to be simplified. One way to simplify a polynomial is to combine the like terms if there are any. Two or more terms in a polynomial are like terms if they have the same variable (or variables) with the same exponent. For example, [latex]3x^{2}[/latex] and [latex]-5x^{2}[/latex] are like terms: They both have x as the variable, and the exponent is 2 for each. However, [latex]3x^{2}[/latex] and [latex]3x[/latex] are not like terms, because their exponents are different.
Here are some examples of terms that are alike and some that are unlike.
| Term | Like Terms | UNLike Terms |
| [latex]a[/latex] | [latex]3a, \,\,\,-2a,\,\,\, \frac{1}{2}a[/latex] | [latex]a^2,\,\,\,\frac{1}{a},\,\,\, \sqrt{a}[/latex] |
| [latex]a^2[/latex] | [latex]-5a^2,\,\,\,\frac{1}{4}a^2,\,\,\, 0.56a^2[/latex] | [latex]\frac{1}{a^2},\,\,\,\sqrt{a^2},\,\,\, a^3[/latex] |
| [latex]ab[/latex] | [latex]7ab,\,\,\,0.23ab,\,\,\,\frac{2}{3}ab,\,\,\,-ab[/latex] | [latex]a^2b,\,\,\,\frac{1}{ab},\,\,\,\sqrt{ab} [/latex] |
| [latex]ab^2[/latex] | [latex]4ab^2,\,\,\, \frac{ab^2}{7},\,\,\,0.4ab^2,\,\,\, -ab^2[/latex] | [latex]a^2b,\,\,\, ab,\,\,\,\sqrt{ab^2},\,\,\,\frac{1}{ab^2}[/latex] |
Example 6
Which of these terms are like terms?
[latex]7x^{3}+7x+7y-8x^{3}+9y-3x^{2}+8y^{2}[/latex]
You can use the distributive property to simplify the sum of like terms. Recall that the distributive property states that the product of a number and a sum (or difference) is equal to the sum (or difference) of the products.
[latex]2\left(3+6\right)=2\left(3\right)+2\left(6\right)[/latex]
Both expressions equal 18. So you can write the expression in whichever form is the most useful.
Let’s see how we can use this property to combine like terms.
Example 7
Simplify [latex]3x^{2}-5x^{2}[/latex].
You may have noticed that combining like terms involves combining the coefficients to find the new coefficient of the like term. You can use this as a shortcut.
Example 8
Simplify [latex]6a^{4}+4a^{4}[/latex].
When you have a polynomial with more terms, you have to be careful that you combine only like terms. If two terms are not like terms, you can’t combine them.
Example 9
Simplify [latex]3x^{2}+3x+x+1+5x[/latex]
Domain of a polynomial function
In Section 3.2, we introduced domain as the set of input values for a function. In the context of a function given by a formula, it is helpful to expand upon this definition, where domain is the set of input values for a function which produce valid input values. It follows given a function [latex]f(x)[/latex], the domain of the function is the set of [latex]x[/latex]-values that produce valid outputs.
Example 10
Let [latex]f(x)=-2x^2+7[/latex].
A. Compute [latex]f(-3)[/latex], [latex]f(0)[/latex], and [latex]f(2)[/latex]. If the answer is undefined, state this.
B. Are each of the values, [latex]-3[/latex], [latex]0[/latex], and [latex]2[/latex], in the domain of [latex]f(x)[/latex]?
C. Give the domain in interval notation.
If we consider the question of domain further, we realize that there was also nothing special about the function given in the previous example in terms of producing valid outputs. It follows that the domain of every polynomial is “all real numbers.”
Summary
Polynomials are algebraic expressions that contain any number of terms combined by using addition or subtraction. A term is a number, a variable, or a product of a number and one or more variables with exponents. Like terms (same variable or variables raised to the same power) can be combined to simplify a polynomial. The polynomials can be evaluated by substituting a given value of the variable into each instance of the variable, then using order of operations to complete the calculations. Lastly, we examined the domain of polynomial functions, revealing a domain of “all real numbers” for all polynomials.