SECTION 4.3 The Real Zeros of a Polynomial Function 215 1 Use the Remainder and Factor Theorems When one polynomial (the dividend) is divided by another (the divisor), a quotient polynomial and a remainder are obtained.The remainder is either the zero polynomial or a polynomial whose degree is less than the degree of the divisor.To check, verify that ( )( ) + = Quotient Divisor Remainder Dividend This checking routine is the basis for a famous theorem called the division algorithm * for polynomials , which we now state without proof. Need to Review? Long division is discussed in Section A.3, pp. A25–A27. REMAINDER THEOREM Suppose f is a polynomial function. If f x( ) is divided by x c, − then the remainder is f c( ) . In equation (1), f x( ) is the dividend , g x( ) is the divisor , q x( ) is the quotient , and r x( ) is the remainder . If the divisor g x( ) is a first-degree polynomial of the form g x x c c a real number ( ) = − then the remainder r x( ) is either the zero polynomial or a polynomial of degree 0. As a result, for such divisors, the remainder is some number, say R , and we may write f x x c q x R ( ) ( ) ( ) = − + (2) This equation is an identity in x and is true for all real numbers x . Suppose that x c = . Then equation (2) becomes ( ) ( ) ( ) ( ) = − + = f c c c q c R f c R Substitute f c( ) for R in equation (2) to obtain f x x c q x f c ( ) ( ) ( ) ( ) = − + (3) which proves the Remainder Theorem . Using the Remainder Theorem Find the remainder when f x x x4 5 3 2 ( ) = − − is divided by (a) x 3 − (b) x 2 + EXAMPLE 1 Solution (a) Either polynomial division or synthetic division could be used, but it is easier to use the Remainder Theorem, which states that the remainder is f 3( ) . f 3 3 43 5 27365 14 3 2 ( ) = − ⋅ − = − − = − The remainder is 14 − . *A systematic process in which certain steps are repeated a finite number of times is called an algorithm . For example, polynomial division is an algorithm. THEOREM Division Algorithm for Polynomials If f x( ) and g x( ) denote polynomial functions and if g x( ) is a polynomial whose degree is greater than zero, then there are unique polynomial functions q x( ) and r x( ) for which f x g x q x r x g x f x q x g x r x or ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = + = + (1) ↑ ↑ ↑ ↑ dividend quotient divisor remainder where r x( ) is either the zero polynomial or a polynomial of degree less than that of g x( ) . (continued)
RkJQdWJsaXNoZXIy NjM5ODQ=