This document defines polynomial functions and discusses their key properties. It defines polynomials as expressions with real number coefficients and positive integer exponents. Examples of polynomials and non-polynomials are provided. The document discusses defining polynomials by degree or number of terms, and classifying specific polynomials. It covers finding zeros of polynomial functions and their multiplicities. The document also addresses end behavior of polynomials based on the leading coefficient and degree. It provides an example of analyzing a polynomial function by defining it, finding zeros and multiplicities, describing end behavior, and sketching its graph.

1. Module 2 Lesson 2
Polynomial Functions:
What is a polynomial function?

2. What is a Polynomial?
A polynomial is expression in the form:
where
 The coefficients (the a values) are real numbers
 The exponents (the n values) are whole numbers (positive
integers)
 The domain is All Real Numbers.
caxxaxaxf n
n
n
n  
 ...)( 1
1

3. Examples
Polynomials NOT Polynomials
xxxxf
xxy


36
3
7
4
3
)(
425
3
5
)(
143 3


x
xf
xxy
Exponents
are not
positive
integers!

4. 𝑓 𝑥 = 5𝑥 + 2𝑥2
− 6𝑥3
+ 3 𝑔 𝑥 = 2𝑥5
− 4𝑥3
+ 𝑥 − 2
ℎ 𝑥 = 2𝑥3(4𝑥5 + 3𝑥)
3 5
8
𝑘 𝑥 = 4𝑥3 + 6𝑥11 − 𝑥10 + 𝑥12
12
State the degree of the following polynomial functions
Ways to define polynomials
 By Degree
 The largest degree of the function is the degree of the polynomial
 By the number of terms.
 Count the number of terms in the expression.

5. Monomial: A number, a variable or the product of a number and one
or more variables.
Polynomial: A monomial or a sum of monomials.
Binomial: A polynomial with exactly two terms.
Trinomial: A polynomial with exactly three terms.
Defining using number of terms
For four or more terms, we just call it a polynomial.

6. .
Polynomial
a.
b.
c.
d.
5
42 x
xx 2
3
14 23
 xx
Degree Classify by degree Classify by number of terms
Zero Constant Monomial
First Linear Binomial
Second Quadratic Binomial
Third Cubic Trinomial
e. 3x4 - 4x3 + 6x2 - 7 Fourth Quartic Polynomial
Classify the polynomials by degree and
number of terms
f. 8x7 - 7x - 9 Seventh Septic or Heptic Trinomial

7. … r is an x-intercept of the graph of the function.
… (x – r) is a factor of the function.
… r is a solution to the function f(x) = 0
If f(r) = 0 and r is a real number, then r is a real zero of the function and….
Solving Polynomial Equations
To solve a polynomial equation you will find the x – intercepts.You find x-intercepts by
letting y = 0 and then using the Zero Product Property (just like when you were
solving quadratics!). Intercepts can be referred to as solutions, roots,or zeros.
The maximum number of solutions a polynomial can
have is limited by the degree of the polynomial!

8. The graph of the function touches the x-axis but does not cross it.
Multiplicities appearing an Even Number of times
To find a Multiplicity
Count the number of times a factor (m) of a function is repeated.
The graph of the function crosses the x-axis.
Multiplicities appearing an Odd Number of times
Multiplicities
Sometimes a solution will appear more than once.This solution has a
multiplicity.

9. 3 is a zero with a multiplicity of
Identify the zeros and their multiplicity
3.-2 is a zero with a multiplicity of
1. Graph crosses the x-axis at x = 3
Graph crosses the x-axis at x = -2.
-4 is a zero with a multiplicity of
2.
1. Graph crosses the x-axis at x = -4.
Graph touches the x-axis at x = 7.
Graph crosses the x-axis at x = -1.
Graph crosses the x-axis at x = 4
2 is a zero with a multiplicity of 2. Graph touches the x-axis at x = 2
7 is a zero with a multiplicity of
-1 is a zero with a multiplicity of 1.
4 is a zero with a multiplicity of 1.

10. If 𝑓 𝑥 = 𝑎𝑥 𝑛 and n is even, then both ends will approach
+.
If 𝑓 𝑥 = −𝑎𝑥 𝑛 and n is even, then both ends will approach –
.
If 𝑓 𝑥 = 𝑎𝑥 𝑛
and n is odd,
then as x  – , 𝑓 𝑥  – and as x , 𝑓 𝑥  .
If 𝑓 𝑥 = −𝑎𝑥 𝑛 and n is odd,
then as x  – , 𝑓 𝑥 and as x , 𝑓 𝑥  –.
End Behavior of a Polynomial
You can predict what directions the ends of the graph
are going based on the sign of the leading coefficient
and the degree of the polynomial.

11. 𝑓 𝑥 = 𝑎𝑥 𝑛
and n is even 𝑓 𝑥 = −𝑎𝑥 𝑛
and n is even
𝑓 𝑥 = 𝑎𝑥 𝑛 and n is odd 𝑓 𝑥 = −𝑎𝑥 𝑛 and n is odd
End Behavior, con’t

12. Time to put it all together!
For the following polynomial:
 Define by number of terms and degree,
 State number of possible solutions,
 Find zeros and state multiplicities,
 Describe multiplicities,
 Describe the end behavior, and
 Sketch graph.
2
)2)(1()(  xxxf

13.  Define by number of terms and degree and state number of possible
solutions
To find the degree and the number of terms, we will need to distribute.
So we have a third degree, or cubic, trinomial.This
trinomial will have a maximum of 3 unique solutions.
2
)2)(1()(  xxxf
43
4444
)44)(1(
)2)(2)(1()(
23
223
2




xx
xxxxx
xxx
xxxxf

14. 2
)2)(1()(  xxxf
21
2010
)2)(1(0 2



xx
xx
xx
• Find zeros and state multiplicities and describe multiplicities.
The solutions to this polynomial are x = 1 and x = -2.
The zero at x =1 has a multiplicity of 1.The graph will cross the x-axis at 1.
The zero at x = -2 has a multiplicity of 2 and will touch the x-axis.

15.  Describe the end behavior and Sketch graph.
Since n = 3, an odd number, we know that the end behavior
will be split- one side will be going up and the other side will
be going down.
As a = +1, the graph will being going up from left to right. So
the left side of the graph is pointing down and the right side of
the graph is pointing up.
43)2)(1()( 232
 xxxxxf

16. Sketch the Graph
1. Plot zeros
2. Choose a point in
between zeros to help
find turning point.
3. Find y-intercept
4. Plot the other points.
5. Use end behavior and
intercepts to graph.
Let x = -1
y = (-2)(-1+2)2 = -2
Let x = 0
y = (-1)(2)2 = -4