Section 6.6 The Fundamental Theorem of Algebra

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Notes 6.6 Fundamental Theorem of Algebra

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Presentation transcript:

Section 6.6 The Fundamental Theorem of Algebra Obj: To use the Fundamental Theorem of Algebra to solve polynomial equations with complex roots

Recall: Complex Numbers = Real #s + Imaginary #s We have solved polynomial equations and found their roots are included in the set of complex numbers. Recall: Complex Numbers = Real #s + Imaginary #s Therefore, our roots have been: -integers -rational numbers -irrational numbers -imaginary numbers But, can all polynomial equations be solved using complex numbers?

Carl Friedrich Gauss (1777-1855) -- says roots of every polynomial equation, even those with imaginary coefficients, are complex numbers. --developed the Fundamental Theorem of Algebra

Fundamental Theorem of Algebra If P(x) is a polynomial of degree n ≥ 1 with complex coefficients, then P(x) = 0 has at least one complex root. Corollary Including imaginary roots and multiple roots, as nth degree polynomial equation has exactly n roots A corollary is a deduction. Another way of saying this is: you can factor a polynomial of degree n into n linear factors.

Imaginary Root Theorem If the imaginary number a +bi is a root of a polynomial equation with real coefficients, then the conjugate a- bi is also a root. Therefore, imaginary roots always _________________________.

Using the Fundamental Theorem of Algebra: Example 1: For the equation x3 + 2x2 – 4x – 6 = 0, find: a. Number of complex roots: b. Number of real roots: c. Possible rational roots:

Example 2: For the equation x4 – 3x3 + x2 – x + 3 = 0, find: a. Number of complex roots: b. Number of real roots: c. Possible rational roots:

Finding all zeros of a Polynomial Function: Example 3: Find the number of complex zeros of f(x) = x5 + 3x4 – x – 3. FIND ALL ZEROS.

f(x) =x3 – 2x2 +4x – 8. FIND ALL ZEROS. Example 4: Find the number of complex zeros of f(x) =x3 – 2x2 +4x – 8. FIND ALL ZEROS.

CLOSURE: Find the degree How can you determine how many roots there are for a polynomial equation? Find the degree

Assignment: Pg 337 # 1-8, 9-15 odd, 21,23