Transcript Chapter 8 Exploring Polynomial Functions

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Exploring Polynomial Functions
Locating Zeros
Graphing Polynomial Functions and
Approximating Zeros
• Look back in Chapter 4 to help with understanding finding
zeros and the definition of even and odd functions
• Location Principle:
– If y = f(x) is a polynomial function and you have a and b such that
f(a) < 0 and f(b) > 0 then there will be some number in between a
and b that is a zero of the function
a
zero
b
• A relative maximum is the highest point between two
zeros and a relative minimum is the lowest point between
two zeros
Let’s use the table function on the graphing
calculators combined with what we know about
possible zeros.
Graph the function f(x) = -2x3 – 5x2 + 3x + 2 and approximate
the real zeros.
There are zeros at
approximately -2.9,
-0.4, and -0.8.
• Upper Bound Theorem
•
If p(x) is divided by x – c and there are
no sign changes in the quotient or
remainder, then c is upper bound
• .
•
Lower Bound Theorem
If p(x) is divided by x + c and there are
alternating sign changes in the quotient and the
remainder, then -c is the lower bound.
Let’s put them to use…
• Find an integral upper
and lower bound of
the zeros of f ( x)  x3  3x 2  5x 10