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In addition, if a polynomial of degree n of real coefficients has n distinct real zeros < < <, we see, using Rolle's theorem, that the zeros of the derivative polynomial are in the interval [,] which is the convex hull of the set of roots.
In particular, the real roots are mostly located near ±1, and, moreover, their expected number is, for a large degree, less than the natural logarithm of the degree. If the coefficients are Gaussian distributed with a mean of zero and variance of σ then the mean density of real roots is given by the Kac formula [21] [22]
Since there are more real numbers than pairs (i, j), one can find distinct real numbers t and s such that z i + z j + tz i z j and z i + z j + sz i z j are complex (for the same i and j). So, both z i + z j and z i z j are complex numbers. It is easy to check that every complex number has a complex square root, thus every complex polynomial of ...
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function, is a member of the domain of such that () vanishes at ; that is, the function attains the value of 0 at , or equivalently, is a solution to the equation () =. [1]
When there is only one distinct root, it can be interpreted as two roots with the same value, called a double root. When there are no real roots, the coefficients can be considered as complex numbers with zero imaginary part, and the quadratic equation still has two complex-valued roots, complex conjugates of each-other with a non-zero ...
All zeros of () are real, distinct from each other, and lie in the interval (,). Furthermore, if ... is a zero of (), so is . These zeros play an ...
If the three roots are real and distinct, the discriminant is a product of positive reals, that is > If only one root, say r 1, is real, then r 2 and r 3 are complex conjugates, which implies that r 2 – r 3 is a purely imaginary number, and thus that (r 2 – r 3) 2 is real and negative.
If P < 0 and D < 0 then all four roots are real and distinct. If P > 0 or D > 0 then there are two pairs of non-real complex conjugate roots. [13] If ∆ = 0 then (and only then) the polynomial has a multiple root. Here are the different cases that can occur: If P < 0 and D < 0 and ∆ 0 ≠ 0, there are a real double root and two real simple ...