Fundamental Theorem of Algebra

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Notes 6.6 Fundamental Theorem of Algebra

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Presentation transcript:

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

OBJECTIVE 1

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

OBJECTIVE 2

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

OBJECTIVE 3

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

f(x) = x4 + 2x3 + x2 - 8x - 20 x = 2 2| 1 2 1 -8 -20 2 8 18 20 _______________ 1 4 9 10 0 Zeros are: 2, -2, -1 + 2i, -1 – 2i x = -2 -2| 1 4 9 10 -2 -4 -10 _______________ 1 2 5 0 f(x) = (x - 2)(x + 2)(x2 +2x + 5) Using quadratic formula for x2 +2x + 5 we get x = -1 + 2i and -1 – 2i