Stitz-Zeager_College_Algebra_e-book

# Recall that the rational zeros theorem required the

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Unformatted text preview: t two zeros. Since there seems to be no other rational zeros to try, we continue with −1. Also, the shape of the crossing at x = −1 leads us to wonder if the zero x = −1 has multiplicity 3. −1 −1 2 4 −1 −6 −3 ↓ −2 −2 3 3 2 2 −3 −3 0 ↓ −2 0 3 2 0 −3 0 Success! Our quotient polynomial is now 2x2 − 3. Setting this to zero gives 2x2 − 3 = 0, or √ 3 x2 = 2 , which gives us x = ± 26 . Concerning multiplicities, based on our division, we have that −1 has a multiplicity of at least 2. The Factor Theorem tells us our remaining zeros, √ ± 26 , each have multiplicity at least 1. However, Theorem 3.7 tells us f can have at most 4 real zeros, counting multiplicity, and so we conclude that −1 is of multiplicity exactly 2 and √ ± 26 each has multiplicity 1. (Thus, we were wrong to think that −1 had multiplicity 3.) It is interesting to note that we could greatly improve on the graph of y = f (x) in the previous example given to us by the calculator. For instance, from√our dete...
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## This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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