Name:
Mods:
4.3 Notes on Polynomial Functions
Types of Polynomial Functions
Degree
Name
0
1
2
3
4
5
Typical Polynomial Graphs:
Cubic Graphs:
a3  0
a3  0
*Note that these graphs can have at most _______ zeros and at most ______ extrema
Quartic Graphs:
a4  0
a4  0
*Note that these graphs can have at most _______ zeros and at most ______ extrema
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End Behavior of Polynomials:
f ( x )  an x n
Example: List the end behavior of the following functions
Function
x  
f ( x)  2 x
f ( x)   x
3
f ( x)  5 x 4
f ( x)  0.4 x 2
Zeros of a Polynomial Function:
Example: Find the zeros of f ( x)  x3  2 x 2  8x
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x 
Multiplicity of Polynomials:
When a factor is repeated, it is called a repeated zero.
Zeros of odd and even multiplicity:
If a polynomial function f has a real zero c of odd multiplicity, then the graph of f crosses the x-axis c.
If a polynomial function f has a real zero c of even multiplicity, then the graph of f does not cross the xaxis at c but will “bounce”
Example: f ( x)  ( x  3)4 ( x  2)3
Example: Find the multiplicity of each of the zeros for the function below, then sketch a graph of the
function g ( x)  ( x  2)3 ( x  1)2
Example: Sketch the graph of a factored polynomial without a calculator
h( x)  ( x  2)3 ( x  4) 4
Homework: 4.3 worksheet
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