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B let c be a real number q which is not in the domain

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Unformatted text preview: b2 is an irreducible quadratic factor of f . As a result, we have our last result of the section. Theorem 3.16. Real Factorization Theorem: Suppose f is a polynomial function with real number coefficients. Then f (x) can be factored into a product of linear factors corresponding to the real zeros of f and irreducible quadratic factors which give the nonreal zeros of f . We now present an example which pulls together all of the major ideas of this section. Example 3.4.3. Let f (x) = x4 + 64. 1. Use synthetic division to show x = 2 + 2i is a zero of f . 2. Find the remaining complex zeros of f . 3. Completely factor f (x) over the complex numbers. 4. Completely factor f (x) over the real numbers. Solution. 1. Remembering to insert the 0’s in the synthetic division tableau we have 2 + 2i 1 0 0 0 64 ↓ 2 + 2i 8i −16 + 16i −64 1 2 + 2i 8i −16 + 16i 0 2. Since f is a fourth degree polynomial, we need to make two successful divisions to get a quadratic quotient. Since 2 + 2i is a zero, we know from Theorem 3.15 that 2 − 2i is also a zero. We continue our synthetic division tableau. 3.4 Complex Zeros and the...
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