1. Polynomial Functions and Models
Section 3.1
Polynomial Functions and Models

2. P(x) = anxn + an-1xn-1 + ... + a1x + a0
Polynomial Functions
A polynomial of degree n is a function of the form
P(x) = anxn + an-1xn a1x + a0
Where an The numbers a0, a1, a2, , an are
called the coefficients of the polynomial. The a0 is
the constant coefficient or constant term. The
number an, the coefficient of the highest power, is
the leading coefficient, and the term anxn is the
leading term.

3. Example of a Polynomial Function

4. Graphs of Polynomial Functions and Nonpolynomial Functions

5. Graphs of Polynomials Graphs are smooth curves Graphs are lines
Degree greater than 2
ex. f(x) = x3
These graphs will not have the following:
Break or hole
Corner or cusp
Graphs are lines
Degree 0 or 1
ex. f(x) = 3 or f(x) = x – 5
Graphs are parabolas
Degree 2
ex. f(x) = x2 + 4x + 8

6. End Behavior of Polynomials
End Behavior- a description of what happens as x becomes large in the positive and negative direction.
End Behavior is determined by:
Term with the highest power of x
Sign of this term’s coefficient

7. Even- and Odd-Degree Functions

8. The Leading-Term Test

9. Finding Zeros of a Polynomial
Zero- another way of saying solution
Zeros of Polynomials
Solutions
Place where graph crosses the x-axis
(x-intercepts)
Zeros of the function
Place where f(x) = 0

10. X-Intercepts (Real Zeros)
A polynomial function of degree n will have at most n x-intercepts (real zeros).

11. Number of Turning Points (relative maxima/minima)
The number of relative maxima/minima of the graph of a polynomial function of degree n is at most n – 1.
ex. f(x) = x4 + 3x3 – 2x2 + 1
Determine number of relative maxima/minima
n – 1 = 4 – 1 = 3

12. Using the Graphing Calculator to Determine Zeros
Graph the following polynomial function and determine the zeros.
Before graphing, determine the end behavior and the number
of relative maxima/minima.
In factored form:
P(x) = (x + 2)(x – 1)(x – 3)²

13. Multiplicity If (x-c)k, k 1, is a factor of a polynomial function P(x) and:
K is even
The graph is tangent to the x-axis at (c, 0)
K is odd
The graph crosses the x-axis at (c, 0)

14. Multiplicity y = (x + 2)²(x − 1)³ Answer.
 −2 is a root of multiplicity 2,
and 1 is a root of multiplicity 3.  
These are the 5 roots:
−2,  −2,  1,  1,  1.

15. Multiplicity y = x³(x + 2)4(x − 3)5 Answer.
0 is a root of multiplicity 3,
-2 is a root of multiplicity 4,
and 3 is a root of multiplicity 5.  

16. True or False? have 1 real zero. 2.) The function has no real zeros.
1.) The function must
have 1 real zero.
2.) The function has no real zeros.
3.) An odd degree polynomial function must have at least 1 real zero.
4.) An even degree polynomial function must have at least 1 real zero.

17. To Graph a Polynomial
Use the leading term to determine the end behavior.
Find all its real zeros (x-intercepts).
Set y = 0.
Use the x-intercepts to divide the graph into intervals and choose a test point in each interval to graph.
Find the y-intercept. Set x = 0.
Use any additional information (i.e. turning points or multiplicity) to graph the function.

19. The Intermediate Value Theorem
Consider a polynomial function P(x) with the points (a, P(a)) and (b, P(b)) on the function.
For any P(x) with real coefficients, suppose that for a ≠ b, P(a) and P(b) are of opposite signs. Then the function has a real zero between a and b.

20. The Intermediate Value Theorem
In other words, if one point is above the x-axis and the other point is below the x-axis, then because P(x) is continuous and will have to cross the x-axis to connect the two points, P(x) must have a zero somewhere between a and b.