POLYNOMIAL, in elementary algebra, an expression com posed of two or more terms combined by operations of addition or subtraction. Thus I-a+ 7b±c and 2 +a— -\/ are poly nomials. A polynomial of two terms is called a binomial, and one of three terms is called a trinomial. An expression consisting of a single term is called a monomial.
The word polynomial is often used with a more technical mean ing, particularly in higher mathematics, to characterize the manner of dependence of an expression on one or more quantities regarded for the time being as independent variables. Importance attaches then to the nature of the operations performed on the variables, rather than to the number of terms, and monomial expressions of suitable form are admitted as special cases. Under this interpretation, which will be adopted throughout the rest of the article, a polynomial in one variable is a sum of terms, each consisting of a power of the variable multiplied by a coefficient independent of the variable, or, as an extreme case, a single such term ; in a polynomial in several variables, each term contains a power of one of the variables or a product of powers of two or more of them. By a power in this connection is meant a power with exponent equal to a positive whole number or zero. The highest exponent that occurs, or, in the case of more than one variable, the highest value attained by the sum of the exponents in a single term, is the degree of the polynomial. Thus is a polynomial of the second degree in x (if aX o), and is a polynomial of the seventh degree in x and y. The latter is also said to be of the fifth degree in x, and of the fourth degree in y.
Having fundamentally a relative sense, the definition is ap plicable to characterize the manner of dependence on quantities which may themselves be more or less complicated expressions of any form. Thus /x)+c, a (log b log x+c are polynomials with respect to x, 1+ r/x, and log x respectively, though the last three are not polynomials with respect to x; and a polynomial in x, can also be regarded as a polynomial in The sum, dif ference, or product of two polynomials is a polynomial ; their quotient in general is not. Polynomials and quotients of poly nomials are known collectively as rational functions.
In elementary algebra, expressions coming under the more technical definition of polynomials are studied largely in con nection with the equations formed by setting them equal to zero. If f(x) denotes the polynomial where ao 0, the equation f(x) = o has n roots, r1, r2, . . . , (which may or may not be all different from each other) ; and then f(x) can be factorized in the form (x—r2) Thus every polynomial of the nth degree in a single variable can be resolved into n factors of the first degree. A corresponding
resolution into factors of the first degree is not possible in general for polynomials in more than one variable. A polynomial (such as which cannot be expressed as a product of polynomials of lower degree is said to be irreducible. Every polynomial which is not itself irreducible can be resolved into irreducible factors, and apart from the order of the factors, and from factors which are merely constant, can be so resolved in only one way.
It is an important fact in trigonometry that the cosine of n times an angle, when n is a whole number, can be expressed as a polynomial of the nth degree in terms of the cosine of the angle itself ; e.g., The sine of nx can be expressed as the product of sin x by a poly nomial of degree n-1 in cos x; when n is odd, but not when n is even, it can also be expressed as a polynomial of the nth degree in sin x; e.g., sin 2X = 2 sin x cos x, sin 3x = sin x (4 cost x— 1)= 3 sin x— 4 x.
The relation between algebra and trigonometry was emphasized by Vieta (154o-16o3), who contributed largely to the advance ment of both branches.
Analytic geometry is largely concerned with the geometric interpretation of the equations obtained by setting polynomials in the co-ordinates equal to zero. In the plane, an equation of the first degree, of the typical form Ax+By-FC=o, represents a straight line; i.e., if (x, y) are the rectangular co-ordinates of a point, all points whose co-ordinates satisfy the equation lie on a straight line, and all points of the line have co-ordinates satis fying the equation. The conic sections (q.v.),—ellipse, circle (which may be regarded as a special case of the ellipse), parabola, hyperbola (qq.v.), and certain "degenerate" forms (pairs of straight lines)—are represented by equations of the second degree, where in particular cases one or more of the coefficients may be equal to zero. Descartes (Geometrie, 1637) and his contemporary, Fermat, are regarded as the founders of analytic geometry (q.v.). The curves represented by equations of the third degree were systematically studied by Newton (1704). In three dimensions, an equation of the first degree, Ax+By+Cz-FD=o, represents a plane, and one of the second degree a quadric surface—ellipsoid, sphere (a special case of the ellipsoid), hyperboloid (of one sheet or of two sheets), paraboloid (elliptic or hyperbolic), cone, cylin der (qq.v.), or, as a degenerate form, a pair of planes.