Fundamental Theorem of Algebra |
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Every polynomial equation with complex coordinates and a degree greater than zero has at least one root in the set of complex numbers. |

A polynomial equation with degree n will have n roots in the set of complex numbers.

Descartes’ Rule of Signs |
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Descartes’ Rule of Signs can be used to determine the number of positive real zeros, negative real zeros, and imaginary zeros in a polynomial function. |

How many zeros (and what kinds of zeros) does this equation have?

Figure 1. |
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After arranging the terms of a polynomial equation into descending powers:

The number of positive real zeros in y = P(x) is equal to the number of changes of sign in front of each term, or is less than this by an even number

and

The number of negative real zeros in y = P(x) is the same as the number of changes of sign in front of the terms of P(-x), or is less than this value by an even number.

First, we test for the number of positive real zeros:

Figure 2. |
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Second, we test for the number of negative real zeros:

Figure 3. |
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So, how many different combinations of zeros (and what kinds of zeros) does this equation have?

Figure 4. |
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