Zero of a function

Every real polynomial of odd degree has an odd number of real roots (counting multiplicities); likewise, a real polynomial of even degree must have an even number of real roots. Consequently, real odd polynomials must have at least one real root (because the smallest odd whole number is 1), whereas even polynomials may have none. This principle can be proven by reference to the intermediate value theorem: since polynomial functions are continuous, the function value must cross zero, in the process of changing from negative to positive or vice versa (which always happens for odd functions).

Computing roots of functions, for example polynomial functions, frequently requires the use of specialised or approximation techniques (e.g., Newton's method). However, some polynomial functions, including all those of degree no greater than 4, can have all their roots expressed algebraically in terms of their coefficients (for more, see algebraic solution).