1. Fundamental Theorem of Algebra
Section 5.3
Complex Zeros;
Fundamental Theorem of Algebra

4. OBJECTIVE 1

6. A polynomial of degree 5 whose coefficients are real numbers has the zeros -2, -3i, and 2+4i. Find the remaining two zeros.
Conjugates: 3i and 2 – 4i

7. OBJECTIVE 2

8. f(x) = a(x + 2)(x – 1)(x – (4 + i))(x – (4 – i))
Find a polynomial f of degree 4 whose coefficients are real numbers and that has the zeros -2, 1, 4+i. Graph the polynomial.
f(x) = a(x + 2)(x – 1)(x – (4 + i))(x – (4 – i))
= (x2 + x – 2)[x2 – x(4 – i) –x(4 + i) + (4 + i)(4 – i)]
= (x2 + x – 2)[(x2 – 4x + ix – 4x – ix + 16 – 4i + 4i – i2]
= (x2 + x – 2)(x2 – 8x + 17)
= x4 – 8x3 + 17x2 + x3 – 8x2 + 17x – 2x2 + 16x - 34
= x4 – 7x3 + 7x2 + 33x - 34

10. OBJECTIVE 3

11. p/q: ±1, ±2, ±4, ±5, ±10, ±20
For leading coefficient of 1

12. f(x) = x4 + 2x3 + x2 - 8x - 20
x = 2 2|
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Zeros are:
2, -2, i, -1 – 2i
x = |
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f(x) = (x - 2)(x + 2)(x2 +2x + 5)
Using quadratic formula for x2 +2x + 5
we get x = i and -1 – 2i