Unformatted text preview: rn to the business of zeros. Suppose we wish to ﬁnd the zeros of f (x) = x2 − 2x + 5.
To solve the equation x2 − 2x + 5 = 0, we note the quadratic doesn’t factor nicely, so we resort to
the Quadratic Formula, Equation 2.5 and obtain
√
−(−2) ± (−2)2 − 4(1)(5)
2 ± −16
2 ± 4i
x=
=
=
= 1 ± 2i.
2(1)
2
2
Two things are important to note. First, the zeros, 1 + 2i and 1 − 2i are complex conjugates.
If ever we obtain nonreal zeros to a quadratic function with real coeﬃcients, the zeros will be a
complex conjugate pair. (Do you see why?) Next, we note that in Example 3.4.1, part 6, we found
(x − [1 + 2i])(x − [1 − 2i]) = x2 − 2x + 5. This demonstrates that the factor theorem holds even for
nonreal zeros, i.e, x = 1 + 2i is a zero of f , and, sure enough, (x − [1 + 2i]) is a factor of f (x). It
turns out that polynomial division works the same way for all complex numbers, real and nonreal
alike, and so the Factor and Remainder Theorems hold as well. But how do we know if...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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