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**Unformatted text preview: **Fundamental Theorem of Algebra 2 + 2i
2 − 2i 225 1
0
0
0
64
↓ 2 + 2i
8i
−16 + 16i −64
1 2 + 2i
8i
−16 + 16i
0
↓ 2 − 2i 8 − 8i 16 − 16i
1
4
8
0 Our quotient polynomial is x2 + 4x + 8. Using the quadratic formula, we obtain the remaining
zeros −2 + 2i and −2 − 2i.
3. Using Theorem 3.14, we get f (x) = (x − [2 − 2i])(x − [2 + 2i])(x − [−2 + 2i])(x − [−2 − 2i]).
4. We multiply the linear factors of f (x) which correspond to complex conjugate pairs. We ﬁnd
(x − [2 − 2i])(x − [2 + 2i]) = x2 − 4x + 8, and (x − [−2 + 2i])(x − [−2 − 2i]) = x2 + 4x + 8.
Our ﬁnal answer f (x) = x2 − 4x + 8 x2 + 4x + 8 .
Our last example turns the tables and asks us to manufacture a polynomial with certain properties
of its graph and zeros.
Example 3.4.4. Find a polynomial p of lowest degree that has integer coeﬃcients and satisﬁes all
of the following criteria:
• the graph of y = p(x) touches the x-axis at 1
3, 0 • x = 3i is a zero of p.
• as x → −∞, p(x) → −∞
• as x → ∞, p(x) → −∞
Solution. To solve this problem, we will need a good understanding of the relationship b...

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