1. 4.2 Polynomial Functions of Higher Degree Objective: Identify zeros and multiplicities, determine end behavior, sketch polynomials

2. What are they? Polynomial functions model many real-world applications
Stock market
Number of new cases in the spread of an epidemic
Objective: Identify real zeros and their multiplicity, end behavior of a polynomial, graph polynomials using zeros, multiplicity, and end behavior

3. A polynomial function is a function of the form
where n is a nonnegative integer and each ai (i = 0, , n) is a real number.
The polynomial function has a leading
coefficient an and degree n.
Examples: Find the leading coefficient and degree of each polynomial function.
Polynomial Function Leading Coefficient Degree
Polynomial Function

4. Graphs of Polynomial Functions
Graphs of polynomial functions are continuous. That is, they have no breaks, holes, or gaps. x y
f (x) = x3 – 5x2 + 4x + 4 x y x y
continuous
not continuous
continuous
smooth
not smooth
polynomial
not polynomial
not polynomial
Polynomial functions are also smooth with rounded turns. Graphs with points or cusps are not graphs of polynomial functions.
Graphs of Polynomial Functions

5. If n is even, their graphs resemble the graph of f (x) = x2.
Polynomial functions of the form f (x) = x n, n  1 are called power functions.
f (x) = x5 x y
f (x) = x4 x y
f (x) = x2
f (x) = x3
If n is even, their graphs resemble the graph of f (x) = x2.
If n is odd, their graphs resemble the graph of f (x) = x3.
Power Functions

6. Graphing
How do we graph polynomial functions when they become really complex like
f(x) = x5 – 2x2 – x + 4
Use x-intercepts (setting f(x) = 0 and solving for x) – this is called finding zeros of a function
Multiplicity of zeros
End behavior

7. Real Zeros of Polynomial Functions
X-intercepts: A real number a is a zero of a function y = f (x) if and only if f (a) = 0.
Real Zeros of Polynomial Functions
If y = f (x) is a polynomial function and a is a real number then the following statements are equivalent.
1. x = a is a zero of f.
2. x = a is a solution of the polynomial equation f (x) = 0.
3. x – a is a factor of the polynomial f (x).
4. (a, 0) is an x-intercept of the graph of y = f (x).
A turning point of a graph of a function is a point at which the graph changes from increasing to decreasing or vice versa.
A polynomial function of degree n has at most n – 1 turning points and at most n zeros.
Zeros of a Function

8. Illustration of 1-4 using f(x) = x2 - 1

9. Example: Find all the real zeros of f (x) = x 4 – x3 – 2x2.
Factor completely: f (x) = x 4 – x3 – 2x2 = x2(x + 1)(x – 2).
The real zeros are x = –1, x = 0, and x = 2. y x
f (x) = x4 – x3 – 2x2
These correspond to the x-intercepts (–1, 0), (0, 0) and (2, 0).
Turning point
The graph shows that there are three turning points. Since the degree is four, this is the maximum number possible.
Turning point
Turning point
Example: Real Zeros

10. You Try
Find all the real zeros of f(x) = x3 + x2 – 2x

11. Repeated Zeros (Multiplicity of a Zero)
If (x – a)n is a factor of a polynomial f, then a is called a zero of
Multiplicity n of f.
1. If n is odd the graph of f (x) crosses the x-axis at (a, 0).
2. If n is even the graph of f (x) touches, but does not cross
through, the x-axis at (a, 0).
Ex: Find the zeros and determine their multiplicity of f (x) = (x – 2)3(x +1)4. x y
Zero Multiplicity Behavior 2 3
odd
crosses x-axis at (2, 0) –1 4
even
touches x-axis at (–1, 0)
Repeated Zeros

12. Examples: Find all the real zeros and state their multiplicities

13. Ex: Find a polynomial…
Find a polynomial of minimum degree whose zeros are: -2 (multiplicity 1), 3 (multiplicity 1) and 1 (multiplicity 2)

14. Leading Coefficient Test
As x grows positively or negatively without bound, the value f (x) of the polynomial function
f (x) = anxn + an – 1xn – 1 + … + a1x + a0 (an  0)
grows positively or negatively without bound depending upon the sign of the leading coefficient an and whether the degree n is odd or even. x y x y
an positive
an negative
n even
n odd
Leading Coefficient Test

15. In a nutshell…If you’re standing at the origin,
degree
Even
Odd
Leading Coefficient
Positive
Negative
“x on the left”
Starts at the top
Starts at the bottom
“x on the right”
Ends at the top
Ends at the bottom
Graph

16. Example: Right-Hand and Left-Hand Behavior
Example: Describe the right-hand and left-hand behavior for the graph of f(x) = –2x3 + 5x2 – x + 1.
Negative -2
Leading Coefficient
Odd 3
Degree x y
f (x) = –2x3 + 5x2 – x + 1
Example: Right-Hand and Left-Hand Behavior

17. Bringing it all together……
For the polynomial f(x) = -(x + 1)2(x – 3)
Determine the zeros, their multiplicity and whether they touch or cross the x-axis.
What is the end behavior?
What is the y-intercept?
Sketch the graph.

18. For the given graph:
Determine the zeros, their SMALLEST multiplicity and whether they
touch or cross the x-axis
Is the degree of the polynomial odd or even?
What is the end behavior?
What is the y-intercept?
Write an equation for the polynomial function.