Unformatted text preview: ared kept the sign of the middle factor the same
on either side of 3. If we look back at the exponents on the factors (x + 2) and x3 , we note they
are both odd - so as we substitute values to the left and right of the corresponding zeros, the signs
of the corresponding factors change which results in the sign of the function value changing. This
is the key to the behavior of the function near the zeros. We need a deﬁnition and then a theorem.
Definition 3.3. Suppose f is a polynomial function and m is a natural number. If (x − c)m is a
factor of f (x) but (x − c)m+1 is not, then we say x = c is a zero of multiplicity m.
Hence, rewriting f (x) = x3 (x − 3)2 (x + 2) as f (x) = (x − 0)3 (x − 3)2 (x − (−2))1 , we see that x = 0
is a zero of multiplicity 3, x = 3 is a zero of multiplicity 2, and x = −2 is a zero of multiplicity 1.
Theorem 3.3. The Role of Multiplicity: Suppose f is a polynomial function and x = c is a
zero of multiplicity m.
• If m is even, the graph of y =...
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