Stitz-Zeager_College_Algebra_e-book

The calculator veries this claim x2 x 2 solving x2 9

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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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This note was uploaded on 05/03/2013 for the course MATH Algebra taught by Professor Wong during the Fall '13 term at Chicago Academy High School.

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