Unformatted text preview: Theorem of Algebra In Section 3.3, we were focused on ﬁnding the real zeros of a polynomial function. In this section, we
expand our horizons and look for the nonreal zeros as well. Consider the polynomial p(x) = x2 + 1.
The zeros of p are the solutions to x2 + 1 = 0, or x2 = −1. This equation has no real solutions, but
you may recall from Intermediate Algebra that we can formally extract the square roots of both
√
√
sides to get x = ± −1. The quantity −1 is usually relabeled i, the socalled imaginary unit.1
The number i, while not a real number, plays along well with real numbers, and acts very much
like any other radical expression. For instance, 3(2i) = 6i, 7i − 3i = 4i, (2 − 7i) + (3 + 4i) = 5 − 3i,
and so forth. The key properties which distinguish i from the real numbers are listed below.
Definition 3.4. The imaginary unit i satisﬁes the two following properties
1. i2 = −1
2. If c is a real number with c ≥ 0 then √ √
−c = i c Property 1 in Deﬁnition 3.4 establishes that i does act as a square root2 of −1, and property 2
establishes what we mean by the ‘principal square root’ of a negative real number. In property
2, it is important to remember the restriction on c. For example, it is perfectly acceptable to say
√
√
√
−4 = i 4 = i(2) = 2i. However, −(−4) = i −4, otherwise, we’d get
2= √ 4= √
...
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 Fall '13
 Wong
 Algebra, Trigonometry, Cartesian Coordinate System, The Land, The Waves, René Descartes, Euclidean geometry

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