1. Polynomial Functions and their Graphs
Section 2.3:
Polynomial Functions and their Graphs
What you’ll learn
Identify polynomial functions.
Recognize characteristics of graphs of polynomial functions.
Determine end behavior.
Use factoring to find zeros of polynomial functions.
Identify zeros and their multiplicities.
Use the intermediate value theorem.
Understand the relationship between degree and turning
points.
8. Graph polynomials functions.

2. Examples of polynomials functions:
Degree 5
Degree 6
Not Polynomial Functions.
The exponent in the variable
is not an integer
The exponent on the variable is not
a nonnegative integer

3. Smooth, Continuous Graphs
End Behavior of Polynomial Functions
The degree of a polynomial function affects the shape of its graph and determines the maximum numbers of turning points. TURNING POINTS are the points where the graphs changes directions. It also affects the end behavior. END BEHAVIOR is the directions of the graph to the far left and the far right.
End behavior of a Polynomial Function of Degree n with Leading Term
n even
n odd
a positive
up and up
down and up
a negative
down and down
up and down

4. Odd-degree polynomial functions have graphs with opposite behavior at each end.
Even-degree polynomial functions have graphs with the same behavior at each end.
Answer
Leading coefficient is 1 (+) and the degree is 3,so is odd
Answer
3+2+1=6 even
Leading coefficient is -4 and the degree is 6,so is even

5. Zeros of Polynomial Functions
Answer
Leading coefficient 1 (+)
Degree 3: odd
Try the G.C.

6. Answer
Make the equation =0

7. Answer Because the multiplicity
of -1 is odd, the graph
crosses the x-axis at
this zero.
Because the multiplicity of 3/2 is even,
the graph touches the x-axis and turns
at this zero.

8. The Intermedia Value Theorem
Use G.C.
use ZERO feature.
Answer
Since f(2)=-1 and f(16), the sign change shows that
the polynomial function has a zero between 2 and 3.
Take a note: If a f is a polynomial function of degree n,
then the graph of f has at most n-1 turning points.

9. Steps to complete a polynomial graph.

10. Answer The leading coefficient,1, is positive,
the degree of the polynomial function,4 is even

11. The partial graph or information we have suggests
y-axis symmetry. Let’s verify this by finding f(-x).
the graph is symmetric with respect to the y-axis.
Because n=4 and 4-1=3
The graphs has three turning points, which is the maximum number for a
fourth degree polynomial function. Verify that 1 is a relative maximum
and(0,1) is turning point.

12. Let’s remember the procedure for dividing one
polynomial by another.

13. Answer _ _ _ _

14. Answer
Take a note: If a power of x is missing in either a dividend or a divisor, add
that power of x with a coefficient of 0 and then divide. In this way, like
terms will be aligned as you carry out the long division. + + +
Remainder

15. Dividing Polynomials Using Synthetic Division.

16. Answer The divisor must be in the form . Thus, we write as .
This means that Writing a 0 coefficient for the missing
Term the dividend, we can express the division as follow

17. The Remainder Theorem. Answer 2 -4 2 1 -2 1 5
By the Remainder Theorem, if f(x) is divide by x-2, then the
remainder is f(2). We’ll use synthetic division to divide.
The remainder,5, is the value of f(2)=5.
We can verify that this is correct by
evaluating f(2) 2 -4 2 1 -2 1 5

18. The Factor Theorem Answer 3 2 -3 -11 6 6 9 -6 2 3 -2 6 9 -6 2 3 -2
The remainder,0, verifies that x-3 is a factor of

19. Using G.C.