350 CHAPTER 3 Polynomial and Rational Functions Carl Friedrich Gauss (1777–1855) The fundamental theorem of algebra was first proved by Carl Friedrich Gauss in his doctoral thesis in 1799, when he was 22 years old. Fundamental Theorem of Algebra Every function defined by a polynomial of degree 1 or more has at least one complex zero. Number of Zeros The fundamental theorem of algebra says that every function defined by a polynomial of degree 1 or more has a zero, which means that every such polynomial can be factored. From the fundamental theorem, if ƒ1x2 is of degree 1 or more, then there is some number k1 such that ƒ1k12 = 0. By the factor theorem, ƒ1x2 = 1x - k12q11x2, for some polynomial q11x2. If q11x2 is of degree 1 or more, the fundamental theorem and the factor theorem can be used to factor q11x2 in the same way. There is some number k2 such that q11k22 = 0, so q11x2 = 1x - k22q21x2 and ƒ1x2 = 1x - k121x - k22q21x2. Assuming that ƒ1x2 has degree n and repeating this process n times gives ƒ1x2 = a1x - k121x - k22 g1x - kn2. a is the leading coefficient. Each of these factors leads to a zero of ƒ1x2, so ƒ1x2 has the n zeros k1, k2, k3, c , kn. This result suggests the number of zeros theorem. Number of ZerosTheorem A function defined by a polynomial of degree n has at most n distinct zeros. NOTE Once we obtained the quadratic factor 6x2 + x - 1 in Example 3, we were able to complete the work by factoring it directly. Had it not been easily factorable, we could have used the quadratic formula to find the other two zeros (and factors). CAUTION The rational zeros theorem gives only possible rational zeros. It does not tell us whether these rational numbers are actual zeros. We must rely on other methods to determine whether or not they are indeed zeros. Furthermore, the polynomial must have integer coefficients. To apply the rational zeros theorem to a polynomial with fractional coefficients, multiply through by the least common denominator of all the fractions. For example, any rational zeros of p1x2 defined below will also be rational zeros of q1x2. p1x2 = x4 - 1 6 x3 + 2 3 x2 - 1 6 x - 1 3 q1x2 = 6x4 - x3 + 4x2 - x - 2 Multiply the terms of p1x2 by 6.
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