1. In the polynomial f(x) = + 2x^4 - 3x^3 + 7x^2 - 8x + 9 we have 4 variations of signs (from 2 to -3, from -3 to 7 etc.), so the possible number of positive real roots is either 4 or less than 4 by an even number, that is either 2 or 0. To estimate the number of negative real roots, consider the function f(-x) = + 2x^4 + 3x^3 + 7x^2 + 8x + 9, where the number of variation of signs is 0. Thus, the first function has no negative real roots.
2. For the polynomial f(x) = 2x^3 + 4x^2 - 7x - 8 there is only one variation of signs, so the number of positive real roots is 1. Consider f(-x) = - 2x^3 + 4x^2 + 7x - 8 that has 2 variations of signs (from -2 to 4 and from 7 to -8). The number of negative real roots is either 2 or 0.
Mar 12th, 2015
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