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In the case of three real roots, the square root expression is an imaginary number; here any real root is expressed by defining the first cube root to be any specific complex cube root of the complex radicand, and by defining the second cube root to be the complex conjugate of the first one.
Every nonzero real number x has exactly one real cube root that is denoted and called the real cube root of x or simply the cube root of x in contexts where complex numbers are not considered. For example, the real cube roots of 8 and −8 are respectively 2 and −2.
In elementary algebra, root rationalisation (or rationalization) is a process by which radicals in the denominator of an algebraic fraction are eliminated.. If the denominator is a monomial in some radical, say , with k < n, rationalisation consists of multiplying the numerator and the denominator by , and replacing by x (this is allowed, as, by definition, a n th root of x is a number that ...
The other roots of the equation are obtained either by changing of cube root or, equivalently, by multiplying the cube root by a primitive cube root of unity, that is . This formula for the roots is always correct except when p = q = 0 , with the proviso that if p = 0 , the square root is chosen so that C ≠ 0 .
whose solutions are called roots of the function. The derivative of a cubic function is a quadratic function. A cubic function with real coefficients has either one or three real roots (which may not be distinct); [1] all odd-degree polynomials with real coefficients have at least one real root.
Moreover, as the number of the real roots is, on the average, proportional to the logarithm of the degree, [2] it is a waste of computer resources to compute the non-real roots when one is interested in real roots. The oldest method for computing the number of real roots, and the number of roots in an interval results from Sturm's theorem, but ...