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Advanced Properties of Polynomials
Intermediate Value Theorem
: If f(x) is a polynomial, and f(a)
≠
f(b) for a<b, then f(x)
takes on ever value from f(a) to f(b) in the closed interval [a,b].
Applied to polynomial zeros, The Intermediate Value Theorem states that if f(a) < 0 and
f(b) > 0, then there must be a value x=c in the interval [a,b] such that f(c) = 0.
In other words, if the graph of a polynomial passes from negative to positive, it
must pass through the xaxis at the value of a zero.
Example:
Given f(x) = x
3
– 2x  1 , f(1) = 2, and f(2) = 3, what does the Intermediate
Value Theorem tell us about the existence of a zero?
f(x) must take on all values from y =  2 to y = 3.
Thus, it must take on the value
y = 0.
This means that there must be a zero at an xvalue between x=1 and x=2.
In fact, there
is a zero, as shown below in the graph.
Fundamental Theorem of Algebra
: If f(x) is a polynomial with degree n, then there is at
least 1 complex zero x=c.
Furthermore, if f(x) has degree n
≥
1 with nonzero leading
coefficient a
n
, then
f(x) has exactly n linear factors and may be written as f(x)= a
n
(x  c
1
)(x  c
2
) . . . (x  c
n
)
where c
1
, c
2
, c
3
, .
...,c
n
are real or complex zeros and some of the zeros and associated
factors may be repeated.
The power on any repeated factor is known as its multiplicity.
Factors that are not repeated have multiplicity = 1.
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 Fall '09
 Crandall
 Anthropology

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