5.1 Polynomial Functions
We’ve discussed linear and exponential functions already. We saw that linear functions have a constant
rate of change while exponential functions may start off slow but quickly speed up. Now we’re going to examine
another category of functions known as polynomials.
Definitions
A polynomial in one variable is any function with a constant multiplied by a variable to a non-negative
(meaning it could be zero), whole number power or the sum or difference of those things. The constant multiplied
by the variable is still known as the coefficient. Let’s look at some examples of polynomials and non-polynomials.
Polynomials
2 D − 3 + 6 C − This polynomial has non-negative exponents.
() 3 − 7
This polynomial has non-negative exponents including
the constant −7 because that is really −7 0.
() −5
This polynomial has non-negative exponents.
Notice that this means a linear function is
officially a polynomial.
Non-Polynomials
() √
This is not a polynomial because of the fraction
exponent for square root.
3
This is not a polynomial because the variable is in the
denominator meaning a negative exponent.
() () 21 + 7
This is not a polynomial because the variable
itself is the power meaning exponentials
are not polynomial functions.
We could also define a polynomial like this:
() ?z z + ?z… z… + ⋯ ? + ? + ?0
In this form we can see that each of the ? values are the coefficients (the subscript is just a notation
meaning the 1st coefficient, the 2nd coefficient, etc.) and that the last term is just a constant (?0 ). A term in a
polynomial is each piece separated by an add or subtract sign. For example, ?z z is a term and ? is another
term in this polynomial definition. The last thing to notice here is that there are a finite number of terms.
Polynomials can’t have infinite terms. There must be a highest exponent value in a polynomial.
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Classification by Degree
That highest exponent value is very useful for us because we classify polynomials by their degree. The
degree of a polynomial is equal to the highest exponent in the polynomial. For example, the polynomial
C 2 is a 3rd degree polynomial because its highest power is the third power. The polynomial
() 2 s − 3 { + 11 is a 7th degree polynomial because its highest power is the seventh power. You might also
have noticed that we typically write polynomials in descending order of their exponents.
Certain degrees of polynomials have specific names. In fact, we already know one. A first degree
polynomial is called linear. For our purposes, we’ll probably only refer to specific names of the first three or four
degrees. Beyond that we would just call it by its degree number. Here’s a chart of the names.
Polynomial Example
Degree Number
Degree Name
0
Constant
1
Linear
2
Quadratic
3
Cubic
4
Quartic
5
5th degree
() {
6
6th degree
…
…
…
() z
<
<th degree
9
2
+ 7
3
() −4 + 2
() C − + 3
() 2 − 5
() −4 D + 2
Again, the fifth degree, sixth degree, etc. all have names, but we won’t be referring to them at this level.
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Classification by Number of Terms
We also can give a polynomial a name based on the number of terms it has. Here’s a chart of the names
that we’ll be using.
Polynomial Example
s
Number of Terms
Term Name
1
Monomial
2
Binomial
3
Trinomial
4
Polynomial
…
…
s 7
() = −4 s + C − 2
() C − + − 1
…
Just like with degree names, a polynomial with four terms or above has a name, but we’ll only be referring
to the first few by name.
Give it a Name!
Now we can name the polynomials by both degree and number of terms. Typically the degree name is used
as an adjective and the number of terms name is the noun. For example:
Polynomial Example
C 7
Cubic Binomial
() = −2 C
Cubic Monomial
() + 3 − 7
Quadratic Trinomial
() − 5
() s + 7 − 1
= D + 7 − 3 + 8
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Name
Quartic Binomial
7th Degree Trinomial
5th Degree Polynomial
Lesson 5.1
Decide whether or not the following functions are polynomials or not. If they are polynomials, name them both
by degree and the number of terms.
1. 1
2. = 4. = √ − 1
5. = s
7. = 31
C
3. = C + + 3
D
s
6. = − 8. = C1
9. = −2 − 8
10. = − 3 + 4
11. = 41 + 2
12. = √ + 3
13. = C − 1
14. = 5 15. = −3 C + 6
16. = − 1
17. = −7 C
18. = 1  ˜1
19. = √ + 5
20. = 2 •
22. = 2 …
23. = −

C

C
21. = C + s
−%
24. = …C + 231