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19 equation contradiction 449 graph 22 identity 449

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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 z+z z−z (c) Show that Re(z ) = and Im(z ) = 2 2i (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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