2     恆等式 (Identities)

2.1   恒等式的義意

等式左方與右方經運算後有完全相同的項(terms)。

例：        證明 (x + 1) (x + 4) = (x + 2) (x + 3) - 2 是恆等式

                解：        左方        =      (x + 1) (x + 4)

                                                =     

                                                =     

                                右方        =      (x + 2) (x + 3) - 2

                                                =     

                                                =     

\         左方 = 右方

\         (x + 1) (x + 4) º (x + 2) (x + 3) - 2

 

等式中的變數(x)，代上任意數都使等式成立。

例：        若A(x + 2) - 4 ≣ 3(x + 1) - B，求A和B的值。

                解：        代x = －2， A(－2 + 2) - 4 =  3(－2 + 1) - B

                                                                        - 4  = －3 – B

                                                                         B    = 1

                                代x = 0，       A(0 + 2) - 4 = 3(0 + 1) - 1

                                                                2A - 4 = 3 - 1

                                                                        2A = 6    

                                                                         A = 3

 

2.2   常用的恆等式 (Common Identities)

1.                              4.    
2.                                  5.    
3.  

 

例：        展開 (5 - 2a)2

解：        (5 - 2a)2  = 52－2(5)(2a)＋(2a)2

                                        = 25－20a＋4a2