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2.2 Notes--Polynomial Functions of Higher Degree
A polynomial is an expression of the form: anxn + an-1xn-1+ …+ a2x2 + a1x + a0
Where the a’s are numbers and the n’s are whole numbers. Note that the degree of each term is
one smaller than the degree of the previous term. Note that an is the _______________________
and a0 is a __________________. The degree of a polynomial is the largest exponent.
Are the following polynomials? If yes, state the degree of the polynomial and the lead coefficient.
1. 5x3 – 4x6 + 7
2. -2(x2 – 5)(x+3)
3. 71/2x + 4
4. 8
5. 1/x2
A polynomial function is a function of the form: f(x) = anxn + an-1xn-1+ …+ a2x2 + a1x + a0
There are two major characteristics of the graphs of polynomial functions:
1. It is continuous (no breaks, jumps, or holes)
2. It has only smooth, rounded turns
The leading coefficient test:
Whether a graph of a polynomial function eventually rises or falls can be determined by the
function’s degree (even or odd) and by the sign of the leading coefficient.
For the polynomial f(x) = anxn + an-1xn-1+ …+ a2x2 + a1x + a0
A. When n is odd:
B. When n is even:
Examples:
Describe the left-hand and right-hand behavior of the graph of each function.
6. f ( x ) 
1 3
x  2x
4
8. f ( x)  7 x5  2 x 4  8x3  5x  2
7. f(x) = -3.5x6 + 5x3 – 1
9. f ( x)  12 x 4  2 x3  5 x  4
Now that we can predict the end behavior of a polynomial function, let’s look closer at the rest of
the graph.
For a polynomial function
With degree “n”:
1. The function has at most n real zeros.
2. The graph has at most n -1 turning points (also called
relative maximums and relative minimums)
What is a zero?
How do you find the zeros?
Zeros = Roots = x-intercepts!!!!
All mean the same thing!!
And, the zeros/roots/x-intercepts of a function
Are the solutions to the related equation (provided
the equation is set equal to 0).
1. Algebraically: set f(x) = 0 and solve for x.
2. Graphically: find the x-intercepts of the graph
Examples:
10. Find all the real zeros of f(x) = -2x4 + 2x2.
11. Find all the real zeros of f(x) = x3 – 12x2 + 36x.
Multiplicities:
The number of times a particular root is used is called its multiplicity.
If a root has an even multiplicity, then it will “bounce” on the graph. If a root has an odd
multiplicity, then it will cross through the x-axis on the graph. If the root has an odd multiplicity
that is larger than 1, then it will cross the graph but also “twist.”
How to graph a polynomial function without using a calculator:
1. Apply the leading coefficient test to determine end behavior.
2. Find the zeros of the polynomial and determine behavior at each zero.
3. Sketch the graph—remember the multiplicities.
12. Sketch a graph of f(x) = x2(x+3)3(x-2) without using a calculator.
13. Sketch the graph of f(x) = 2x3 – 6x2 without using a calculator.
14. Sketch the graph of f(x) = 4x3 – 12x2 + 9x without using a calculator.
Practice Problems: pg. 148 #1-8 (all), 14-22 (all), 67,68,71,73,77,79,80 (you can ignore their
instructions on 67-80 and just make a rough sketch like we did on #12-14 of the notes.