上一篇最尾講到，x 2 + 6 x + 9 x^2 + 6x + 9 x 2 + 6 x + 9 x 2 − 9 x^2 - 9 x 2 − 9
恆等式（identity）：無論代乜嘢數入去，兩邊都一樣嘅等式。1. 方程（equation）淨係喺某啲值先成立，例如 2 x = 6 2x = 6 2 x = 6 x = 3 x = 3 x = 3 ( x + 3 ) 2 = x 2 + 6 x + 9 (x + 3)^2 = x^2 + 6x + 9 ( x + 3 ) 2 = x 2 + 6 x + 9 完全平方：中間項係 2ab 完全平方（perfect square）恆等式：a 2 + 2 a b + b 2 = ( a + b ) 2 a^2 + 2ab + b^2 = (a + b)^2 a 2 + 2 ab + b 2 = ( a + b ) 2 a 2 − 2 a b + b 2 = ( a − b ) 2 a^2 - 2ab + b^2 = (a - b)^2 a 2 − 2 ab + b 2 = ( a − b ) 2 2 a b 2ab 2 ab 點解會係咁？用返我哋展開嗰陣用開嘅長方形面積嚟睇。x 2 + 6 x + 9 x^2 + 6x + 9 x 2 + 6 x + 9 x + 3 x + 3 x + 3 x 2 x^2 x 2 3 x 3x 3 x 6 x 6x 6 x 2 a b 2ab 2 ab a b ab ab
認完全平方要三樣一齊驗。例如 9 x 2 − 12 x + 4 9x^2 - 12x + 4 9 x 2 − 12 x + 4 9 x 2 = ( 3 x ) 2 9x^2 = (3x)^2 9 x 2 = ( 3 x ) 2 4 = 2 2 4 = 2^2 4 = 2 2 2 × 3 x × 2 = 12 x 2 \times 3x \times 2 = 12x 2 × 3 x × 2 = 12 x 9 x 2 − 12 x + 4 = ( 3 x − 2 ) 2 9x^2 - 12x + 4 = (3x - 2)^2 9 x 2 − 12 x + 4 = ( 3 x − 2 ) 2 x 2 + 5 x + 9 x^2 + 5x + 9 x 2 + 5 x + 9 2 × x × 3 = 6 x 2 \times x \times 3 = 6x 2 × x × 3 = 6 x 5 x 5x 5 x x 2 + 6 x − 9 x^2 + 6x - 9 x 2 + 6 x − 9
平方差：兩個平方相減 平方差（difference of two squares）恆等式：a 2 − b 2 = ( a + b ) ( a − b ) a^2 - b^2 = (a + b)(a - b) a 2 − b 2 = ( a + b ) ( a − b ) a 2 + b 2 a^2 + b^2 a 2 + b 2 點解 x 2 − 9 x^2 - 9 x 2 − 9 ( x + 3 ) ( x − 3 ) (x + 3)(x - 3) ( x + 3 ) ( x − 3 ) x x x 3 × 3 3 \times 3 3 × 3 x 2 − 9 x^2 - 9 x 2 − 9 x + 3 x + 3 x + 3 x − 3 x - 3 x − 3
用嘅時候，先將每一項寫成平方。例如 4 x 2 − 25 = ( 2 x ) 2 − 5 2 = ( 2 x + 5 ) ( 2 x − 5 ) 4x^2 - 25 = (2x)^2 - 5^2 = (2x + 5)(2x - 5) 4 x 2 − 25 = ( 2 x ) 2 − 5 2 = ( 2 x + 5 ) ( 2 x − 5 ) ( x + 5 ) ( x − 5 ) (x + 5)(x - 5) ( x + 5 ) ( x − 5 ) x 2 − 25 x^2 - 25 x 2 − 25 x 4 − 16 = ( x 2 + 4 ) ( x 2 − 4 ) = ( x 2 + 4 ) ( x + 2 ) ( x − 2 ) x^4 - 16 = (x^2 + 4)(x^2 - 4) = (x^2 + 4)(x + 2)(x - 2) x 4 − 16 = ( x 2 + 4 ) ( x 2 − 4 ) = ( x 2 + 4 ) ( x + 2 ) ( x − 2 ) x 2 − 4 x^2 - 4 x 2 − 4 x 2 + 4 x^2 + 4 x 2 + 4
立方和與立方差：記住「同、異、正」 立方和（sum of two cubes）同立方差（difference of two cubes）恆等式：a 3 + b 3 = ( a + b ) ( a 2 − a b + b 2 ) a^3 + b^3 = (a + b)(a^2 - ab + b^2) a 3 + b 3 = ( a + b ) ( a 2 − ab + b 2 ) a 3 − b 3 = ( a − b ) ( a 2 + a b + b 2 ) a^3 - b^3 = (a - b)(a^2 + ab + b^2) a 3 − b 3 = ( a − b ) ( a 2 + ab + b 2 ) a b ab ab b 2 b^2 b 2 呢兩條唔使死記，展開一次就明。以立方差為例：
( a − b ) ( a 2 + a b + b 2 ) = a 3 + a 2 b + a b 2 − a 2 b − a b 2 − b 3 = a 3 − b 3 (a - b)(a^2 + ab + b^2) = a^3 + a^2 b + ab^2 - a^2 b - ab^2 - b^3 = a^3 - b^3 ( a − b ) ( a 2 + ab + b 2 ) = a 3 + a 2 b + a b 2 − a 2 b − a b 2 − b 3 = a 3 − b 3 中間四項兩兩抵銷，淨低頭尾。所以第二個括號中間係 a b ab ab 2 a b 2ab 2 ab 2 a b 2ab 2 ab a 2 a^2 a 2 b 2 b^2 b 2 2 a b 2ab 2 ab
例題：先抽公因式，再認立方 因式分解 16 x 3 − 54 16x^3 - 54 16 x 3 − 54
先睇有冇公因式：16 同 54 都係雙數，抽出 2：16 x 3 − 54 = 2 ( 8 x 3 − 27 ) 16x^3 - 54 = 2(8x^3 - 27) 16 x 3 − 54 = 2 ( 8 x 3 − 27 ) 將括號入面兩項寫成立方：8 x 3 = ( 2 x ) 3 8x^3 = (2x)^3 8 x 3 = ( 2 x ) 3 27 = 3 3 27 = 3^3 27 = 3 3 a = 2 x a = 2x a = 2 x b = 3 b = 3 b = 3 套立方差恆等式，第一個括號跟原式用減號：( 2 x ) 3 − 3 3 = ( 2 x − 3 ) [ ( 2 x ) 2 + ( 2 x ) ( 3 ) + 3 2 ] (2x)^3 - 3^3 = (2x - 3)\left[(2x)^2 + (2x)(3) + 3^2\right] ( 2 x ) 3 − 3 3 = ( 2 x − 3 ) [ ( 2 x ) 2 + ( 2 x ) ( 3 ) + 3 2 ] 逐項計第二個括號，中間嗰項係 a b = 6 x ab = 6x ab = 6 x 12 x 12x 12 x ( 2 x ) 2 + ( 2 x ) ( 3 ) + 3 2 = 4 x 2 + 6 x + 9 (2x)^2 + (2x)(3) + 3^2 = 4x^2 + 6x + 9 ( 2 x ) 2 + ( 2 x ) ( 3 ) + 3 2 = 4 x 2 + 6 x + 9 記得前面個 2：16 x 3 − 54 = 2 ( 2 x − 3 ) ( 4 x 2 + 6 x + 9 ) 16x^3 - 54 = 2(2x - 3)(4x^2 + 6x + 9) 16 x 3 − 54 = 2 ( 2 x − 3 ) ( 4 x 2 + 6 x + 9 ) 驗算：代 x = 1 x = 1 x = 1 16 − 54 = − 38 16 - 54 = -38 16 − 54 = − 38 2 ( 2 − 3 ) ( 4 + 6 + 9 ) = 2 × ( − 1 ) × 19 = − 38 2(2 - 3)(4 + 6 + 9) = 2 \times (-1) \times 19 = -38 2 ( 2 − 3 ) ( 4 + 6 + 9 ) = 2 × ( − 1 ) × 19 = − 38 4 x 2 + 6 x + 9 4x^2 + 6x + 9 4 x 2 + 6 x + 9 ( 2 x ) 2 (2x)^2 ( 2 x ) 2 3 2 3^2 3 2 12 x 12x 12 x 6 x 6x 6 x
總結：見到一條式，先抽公因式，再數有幾多項。兩項就睇吓係咪平方差、立方和或者立方差；三項就驗吓係咪完全平方，頭尾係平方、中間係 2 a b 2ab 2 ab x 2 − 9 x 2 + 6 x + 9 \frac{x^2 - 9}{x^2 + 6x + 9} x 2 + 6 x + 9 x 2 − 9 ( x + 3 ) ( x − 3 ) (x + 3)(x - 3) ( x + 3 ) ( x − 3 ) ( x + 3 ) 2 (x + 3)^2 ( x + 3 ) 2 x + 3 x + 3 x + 3 x − 3 x + 3 \frac{x - 3}{x + 3} x + 3 x − 3
詳情：DSE 數學補習課程
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