College Algebra
Chapter 5: Polynomial Functions
5.4: The Fundamental Theorem of Algebra
Fundamental Theorem of Algebra – If f is a polynomial of degree n, with n ≥ 1, then f has at
least one zero (at least one root, at least one solution). The solution may be a non-real complex
number.
The Linear Factors Theorem – an nth degree polynomial can be factored as a product of
n linear factors.
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Multiple Zeros -
Geometric Meaning of Multiplicity – In general, if c is a real zero of multiplicity k of a
polynomial f (if (x – c)k is a factor of f), the graph of f will touch the x-axis at (c, 0) and:
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Determine the degree, the y-intercept, the x-intercept(s), and the zero(s) of ‘f’ at which it
“flattens out.” Then graph.
Ex. 1:
**Think . . . End Behavior:
y-intercept:
Ex. 2: Use all available methods to find the zeros of the following polynomial function
The Conjugate Roots Theorem – If the complex number a + bi is a zero of f, then so is the complex
number a – bi. In terms of the linear factors of f, this means that if x – (a + bi) is a factor of f,
then so is x – (a – bi).
Ex. 3: Given that 4 – 3i is a zero of a polynomial 𝒇(𝒙) = 𝒙𝟒 − 𝟖𝒙𝟑 + 𝟐𝟎𝟎𝒙 − 𝟔𝟐𝟓, factor
f completely.
Ex. 4: Construct a 4th degree real-coefficient polynomial function f with zeros of 2, -5, and 1 + i
such that f(1) = 12.
Ex. 5: Construct a polynomial function with the following properties:
fifth degree; 3 is a zero of multiplicity of 3; -2 is the only other zero; leading coefficient is 2