11.1. Fundamental Theorem of Algebra http://www.ck12.org
11.1 Fundamental Theorem of Algebra
Here you will state the connection between zeroes of a polynomial and the Fundamental Theorem of Algebra and
start to use complex numbers.
You have learned that a quadratic has at most two real zeroes and a cubic has at most three real zeros. You may
have noticed that the number of real zeros is always less than or equal to the degree of the polynomial. By looking
at a graph you can see when a parabola crosses thexaxis 0, 1 or 2 times, but what does this have to do with complex
numbers?
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MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/62125http://www.youtube.com/watch?v=NeTRNpBI17I James Sousa: Complex Numbers
Guidance
A real number is any rational or irrational number. When a real number is squared, it will always produce a non-
negative value. Complex numbers include real numbers and another type of number called imaginary numbers.
Unlike real numbers, imaginary numbers produce a negative value when squared. The square root of negative one
is defined to be the imaginary numberi.
i=√−1 andi^2 =− 1
Complex numbers are written with a real component and an imaginary component. All complex numbers can be
written in the forma+bi. When the imaginary component is zero, the number is simply a real number. This means
that real numbers are a subset of complex numbers.
TheFundamental Theorem of Algebrastates that annthdegree polynomial with real or complex coefficients has,
with multiplicity, exactlyncomplex roots. This means a cubic will have exactly 3 roots, some of which may be
complex.
Multiplicity refers to when a root counts more than once. For example, in the following function the only root
occurs atx=3.
f(x) = (x− 3 )^2
The Fundamental Theorem of Algebra states that this 2nddegree polynomial must have exactly 2 roots with multi-
plicity. This means that the rootx=3 has multiplicity 2. One way to determine the multiplicity is to simply look
at the degree of each of the factors in the factorized polynomial.
g(x) = (x− 1 )(x− 3 )^4 (x+ 2 )
This function has 6 roots. The first two rootsx=1 andx=−2 have multiplicities of 1 because the power of each
of their binomial factors is 1. The third rootx=3 has a multiplicity of 4 because the power of its binomial factor is