In elementary algebra letters are used to stand for numbers. For example, in the equation ax2+bx+c=0, the letters a, b, and c stand for various known constant numbers called coefficients and the letter x is an unknown variable number whose value depends on the values of a, b, and c and may be determined by solving the equation. Much of classical algebra is concerned with finding solutions to equations or systems of equations, i.e., finding the roots, or values of the unknowns, that upon substitution into the original equation will make it a numerical identity. For example, x=-2 is a root of x2-2x-8=0 because (-2)2-2(-2)-8=4+4-8=0; substitution will verify that x=4 is also a root of this equation.

The equations of elementary algebra usually involve polynomial functions of one or more variables (see function). The equation in the preceding example involves a polynomial of second degree in the single variable x (see quadratic). One method of finding the zeros of the polynomial function f(x), i.e., the roots of the equation f(x)=0, is to factor the polynomial, if possible. The polynomial x2-2x-8 has factors (x+2) and (x-4), since (x+2)(x-4)=x2-2x-8, so that setting either of these factors equal to zero will make the polynomial zero. In general, if (x-r) is a factor of a polynomial f(x), then r is a zero of the polynomial and a root of the equation f(x)=0. To determine if (x-r) is a factor, divide it into f(x); according to the Factor Theorem, if the remainder f(r)—found by substituting r for x in the original polynomial—is zero, then (x-r) is a factor of f(x). Although a polynomial has real coefficients, its roots may not be real numbers; e.g., x2-9 separates into (x+3)(x-3), which yields two zeros, x=-3 and x=+3, but the zeros of x2+9 are imaginary numbers.

The Fundamental Theorem of Algebra states that every polynomial f(x)=anxn+an-1xn-1+ … +a1x+a0, with an≠0 and n≥1, has at least one complex root, from which it follows that the equation f(x)=0 has exactly n roots, which may be real or complex and may not all be distinct. For example, the equation x4+4x3+5x2+4x+4=0 has four roots, but two are identical and the other two are complex; the factors of the polynomial are (x+2)(x+2)(x+i)(x-i), as can be verified by multiplication.

Modern algebra is yet a further generalization of arithmetic than is classical algebra. It deals with operations that are not necessarily those of arithmetic and that apply to elements that are not necessarily numbers. The elements are members of a set and are classed as a group, a ring, or a field according to the axioms that are satisfied under the particular operations defined for the elements. Among the important concepts of modern algebra are those of a matrix and of a vector space.