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Unformatted text preview: rst two factors together and then the second two factors together, thus
pairing up the complex conjugate pairs of zeros Theorem 3.15 told us we’d get, we have
that p(x) = x2 − 2x + 2)(x2 + 2x + 2). Use the 12 complex 12th roots of 4096 to factor
p(x) = x12 − 4096 into a product of linear and irreducible quadratic factors. 856 Applications of Trigonometry 8. Complete the proof of Theorem 11.14 by showing that if w = 0 than 1
w = 1
|w | . 9. Recall from Section 3.4 that given a complex number z = a + bi its complex conjugate, denoted
z , is given by z = a − bi.
(a) Prove that |z | = |z |.
(b) Prove that |z | = z z
(c) Show that Re(z ) =
and Im(z ) =
(d) Show that if θ ∈ arg(z ) then −θ ∈ arg (z ). Interpret this result geometrically.
(e) Is it always true that Arg (z ) = −Arg(z )?
10. Given an natural number n with n ≥ 2, the n complex nth roots of the number z = 1 are
called the nth Roots of Unity. In the following exercises, assume that n is a ﬁxed, but
arbitrary, natural number such that n ≥ 2.
(a) Show that w = 1 is an nth root of unity.
(b) Show that if both wj and wk are nth roots of unity then so is their product wj wk .
(c) Show that if wj is an nth root of unity then there exists another nth root of unity wj
such that wj wj = 1. Hint: If wj = cis(θ) let wj = cis(2π − θ). You’ll need to verify
that wj = cis(2π − θ) is indeed an nth root of unity.
11. Another way to express the polar form of a complex number is...
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