If a function f(x) has the value 0 for x = a, a is a root of the equation f(x) = 0. The fundamental theorem of algebra states that any algebraic equation of the form a0xn + a1xn−1 + ··· an−1x + an = 0, where the ak's are real numbers, has at least one root. From this it follows readily that such an equation has roots, real or complex, in number equal to the index (here n) of the highest power of x.
Furthermore, if a + ib (where i = Ö−1) is a complex root of the given equation, so is a − ib, the conjugate of a + ib. Equations of degrees up to four may be solved algebraically. This statement means that the roots may be expressed as functions of the coefficients, the functions involving the elementary arithmetical processes of addition, multiplication, raising a number to a power, or extracting the root of a certain order of a given number. It was proved by H. Abel and by E. Galois that it is not possible to solve algebraically the general algebraic equation of degree higher than four. However, it is possible to determine the real roots of an algebraic equation to any desired degree of approximation. See also Calculus; Equations, theory of;



. The
.

