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For n = 1, the cyclotomic polynomial is Φ 1 (x) = x − 1 Therefore, the only primitive first root of unity is 1, which is a non-primitive n th root of unity for every n > 1. As Φ 2 (x) = x + 1, the only primitive second (square) root of unity is −1, which is also a non-primitive n th root of unity for every even n > 2.
For example, −2 has a real 5th root, = … but −2 does not have any real 6th roots. Every non-zero number x, real or complex, has n different complex number nth roots. (In the case x is real, this count includes any real nth roots.) The only complex root of 0 is 0.
An example of a more complicated (although small enough to be written here) solution is the unique real root of x 5 − 5x + 12 = 0. Let a = √ 2φ −1, b = √ 2φ, and c = 4 √ 5, where φ = 1+ √ 5 / 2 is the golden ratio. Then the only real solution x = −1.84208... is given by
The subtraction of only multiples of 2 from the maximal number of positive roots occurs because the polynomial may have nonreal roots, which always come in pairs since the rule applies to polynomials whose coefficients are real. Thus if the polynomial is known to have all real roots, this rule allows one to find the exact number of positive and ...
Quintic function: Fifth degree polynomial. Rational functions: A ratio of two polynomials. nth root. Square root: Yields a number whose square is the given one. Cube root: Yields a number whose cube is the given one.
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For the polynomial f(x) = x 5 − x − 1, the lone real root x = 1.1673... is algebraic, but not expressible in terms of radicals. The other four roots are complex numbers. Van der Waerden [11] cites the polynomial f(x) = x 5 − x − 1. By the rational root theorem, this has no rational zeroes. Neither does it have linear factors modulo 2 or 3.
Real-root isolation is useful because usual root-finding algorithms for computing the real roots of a polynomial may produce some real roots, but, cannot generally certify having found all real roots. In particular, if such an algorithm does not find any root, one does not know whether it is because there is no real root.