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Because x2 1 0 for all real numbers x the fraction x21

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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 non-real 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 re-labeled i, the so-called 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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