Question 764641: How could you use Descartes' rule and the Fundamental Theorem of Algebra to predict the number of complex roots to a polynomial as well as find the number of possible positive and negative real roots to a polynomial?
Answer by rothauserc(4718)   (Show Source):
You can put this solution on YOUR website! Descartes' rule states that the possible number of the positive roots of a polynomial is equal to the number of sign changes in the coefficients of the terms or less than the sign changes by a multiple of 2.
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The Fundamental Theorem of Algebra states that every polynomial equation over the field of complex numbers of degree higher than 1 has a complex solution, furthermore any polynomial of degree n has n roots.
Remember that the complex numbers include the real numbers.
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Suppose we are given the polynomial x^3+3x^2-x-x^4-2, we arrange the terms of the polynomial in the descending order of exponents:
-x^4+x^3+3x^2-x-2, count the number of sign changes, there are 2 sign changes in the polynomial, so the possible number of positive roots of the polynomial is 2 or 0.
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returning to our polynomial above, -x^4+x^3+3x^2-x-2, it has degree 4 and so has n roots. Note that complex roots always come in pairs, so here is what can be said from these two rules:
degree 1 has 1 real root
degree 2 has 2 real roots or 2 complex roots
degree 3 has 3 real roots or 1 real root and 2 complex roots
degree 4 has 4 real roots or 2 real roots and 2 complex roots
note that if the degree is odd, there will be at least 1 real root
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