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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.
1. Remembering to insert the 0’s in the synthetic division tableau we have
2 + 2i 1
↓ 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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