216 CHAPTER 4 Polynomial and Rational Functions (b) To find the remainder when f x( ) is divided by x x 2 2 , ( ) + = − − find f 2 ( ) − . f 2 2 42 5 8165 29 3 2 ( ) ( ) ( ) −=− −− −=−− −=− The remainder is 29 − . COMMENT A graphing utility provides another way to find the value of a function using the eVALUEate feature. Consult your manual for details. See Figure 26 for the result of Example 1(a). ■ Figure 26 Value of a function on a TI-84 Plus CE 10 250 23 5 FACTOR THEOREM Suppose f is a polynomial function. Then x c − is a factor of f x( ) if and only if f c 0 ( ) = . • If f c 0, ( ) = then x c − is a factor of f x( ) . • If x c − is a factor of f x( ) , then f c 0 ( ) = . Compare the method used in Example 1(a) with the method used in Example 1 of Section A.4. Which method do you prefer? An important and useful consequence of the Remainder Theorem is the Factor Theorem . Proof • Suppose that f c 0 ( ) = . Then, by equation (3), we have f x x c q x ( ) ( ) ( ) = − for some polynomial q x( ) . That is, x c − is a factor of f x( ) . • Suppose that x c − is a factor of f x( ) . Then there is a polynomial function q for which f x x c q x ( ) ( ) ( ) = − Replacing x by c , we find that f c c c q c q c 0 0 ( ) ( ) ( ) ( ) = − = ⋅ = This completes the proof. ■ Using the Factor Theorem Use the Factor Theorem to determine whether the function f x x x x 2 2 3 3 2 ( ) = − + − has the factor (a) x 1 − (b) x 2 + EXAMPLE 2 Solution The Factor Theorem states that if f c 0, ( ) = then x c − is a factor. (a) Because x 1 − is of the form x c − with c 1, = find the value of f (1). f 1 21 1 213 2123 0 3 2 ( ) = ⋅ − + ⋅ − = − + − = Figure 27(a) shows the graph of f using a TI-84 Plus CE graphing calculator. Because f 1 0 ( ) = , by the Factor Theorem, x 1 − is a factor of f x( ) . (b) To test the factor x 2, + first write it in the form x c − . Since x x 2 2 , ( ) + = − − find the value of f 2 ( ) − . (a) Figure 27 40 240 25 5 One use of the Factor Theorem is to determine whether a polynomial has a particular factor. The Factor Theorem actually consists of two separate statements: The proof requires two parts.
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