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**Unformatted text preview: **force multiply using the distributive property and see that
(x − [1 + 2i])(x − [1 − 2i]) = x2 − x[1 − 2i] − x[1 + 2i] + [1 − 2i][1 + 2i]
= x2 − x + 2ix − x − 2ix + 1 − 2i + 2i − 4i2
= x2 − 2x + 5 A couple of remarks about the last example are in order. First, the conjugate of a complex number
a + bi is the number a − bi. The notation commonly used for conjugation is a ‘bar’: a + bi = a − bi.
√
√
For example, 3 + 2i = 3 − 2i, 3 − 2i = 3 + 2i, 6 = 6, 4i = −4i, and 3 + 5 = 3 + 5. The properties
of the conjugate are summarized in the following theorem.
5 We will talk more about this in a moment. 3.4 Complex Zeros and the Fundamental Theorem of Algebra 221 Theorem 3.12. Suppose z and w are complex numbers.
• z=z
• z+w =z+w
• z w = zw
• (z )n = z n , for any natural number n = 1, 2, 3, . . .
• z is a real number if and only if z = z .
Essentially, Theorem 3.12 says that complex conjugation works well with addition, multiplication,
and powers. The proof of these properties can best be ac...

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