5.7 The Fundamental Theorem of Algebra Name: ________________
Objectives: Students will be able to use the FTA to solve polynomial
equations with complex solutions.
The equation x3 - 5x2 - 8x - 48 = 0, which becomes
(x+3)(x-4)2 = 0 when factored, has only two distinct
solutions: -3 and 4. Because the factor x-4 appears
twice, however, you can count the solution 4 twice.
So, with 4 counted as a repeated solution, this third
degree equation has three solutions : -3,4 and 4.
This result is generalized as The Fundamental Theorem of
Algebra.
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Fundamental Theorem of Algebra
If f(x) is a polynomial of
degree n where n > 0,
then the equation f(x) = 0
has at least one solution
in the set of complex numbers.
Karl Friedrich
Gauss (1777-1855)
Corollary: If f(x) is a polynomial of degree
n where n > 0, then the equation f(x) = 0 has
exactly n solutions provided each solution
repeated twice is counted as two solutions,
each solution repeated three times is
counted as three solutions, etc.
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Examples:
1.) How many complex solutions does the equation x4 + 5x2 - 36 = 0
have?
2.) How many zeros does the function f(x) = x3 + 7x2 + 8x - 16 have?
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Examples: Find all zeros of the polynomial function.
1.) f(x) = x4 - 6x3 + 7x2 + 6x - 8
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2.) g(x) = x4 - 9x2 - 4x + 12
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Examples: Find all zeros of the polynomial function.
1.) f(x) = x4 + 15x2 - 16
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2.) g(x) = 4x4 + 4x3 - 11x2 - 12x - 3
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