Stitz-Zeager_College_Algebra_e-book

# We formalize the concepts of vertical and horizontal

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Unformatted text preview: can ﬁnd the remaining zeros using the Quadratic Formula, if necessary. Using the techniques developed in Section 3.3, we get 3.4 Complex Zeros and the Fundamental Theorem of Algebra 1 2 1 2 −1 3 223 12 −20 19 −6 −2 1 ↓ 6 −7 6 0 −1 12 −14 12 0 −2 0 ↓ 6 −4 4 2 12 −8 8 40 ↓ −4 4 −4 12 −12 12 0 √ i Our quotient is 12x2 − 12x + 12, whose zeros we ﬁnd to be 1±2 3 . From Theorem 3.14, we know f has exactly 5 zeros, counting multiplicities, and as such we have the zero 1 with 2 1 multiplicity 2, and the zeros − 3 , √ 1+i 3 2 and √ 1−i 3 2, each of multiplicity 1. 2. Applying Theorem 3.14, we are guaranteed that f factors as 1 f (x) = 12 x − 2 2 1 x+ 3 √ 1+i 3 x− 2 √ 1−i 3 x− 2 A true test of Theorem 3.14 (and a student’s mettle!) would be to take the factored form of f (x) in the previous example and multiply it out8 to see that it really does reduce to the formula f (x) = 12x5 − 20x4 + 19x3 − 6x2 − 2x + 1. When factoring a polynomial using Theorem 3.14,...
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