中三數學~因式分解(20點)

a)x^3 + 27
= x^3 + 3^3
= (x+3)(x^2–3x+3^2) -->利用原式: a^3+b^3=(a+ b)(a^2–ab+b^2)
= (x+3)(x^2–3x+9)

b)x^3–7x+ 6
= x^3–7x+(27–21)-->因為27–21=6
= x^3+27–7x–21
= (x^3+27)–7x–21
= (x+3)(x^2–3x+ 9)–7x–21-->利用a的答案
= (x+3)(x^2–3x+ 9)–7(x+3)-->再抽-7出來
= (x+3)[(x^2–3x+ 9)-7]-->再抽(x+3)出來
= (x+3)(x^2–3x+ 9-7)-->拆括號
= (x+3)(x^2–3x+2)
= (x+3)(x-2)(x-1)-->再將(x^2–3x+2)化簡

這便是最後的答案(x+3)(x-2)(x-1)

***做這一類數並不是太複習的!***
因為很多時b部分是利用a的答案的
所以只要小心化解a部分, b的答案便會容易找到的
當然化解的同時, 一定要小心正負數, 拆括號等問題

x^3 + 27
=(x^3) + (3^3)
=(x + 3)(x^2 - 3x + 9)


由此x^3 - 7x + 6
=x^3 - 7x + 6 + 21 - 21
=x^3 - 7x + 27 - 21
=x^3 + 27 - 7x - 21
=(x + 3)(x^2 - 3x + 9) - 7(x + 3)
=(x + 3)[(x^2 -3x + 9 - 7)
=(x + 3)(x^2 - 3x + 2) ＜－ 十字相乘
=(x + 3)(x - 2)(x - 1) ＜－最後答案

1)x^3 + 27
=x^3 + 3^3
=(x + 3)(x^2 - 3x + 9)



2)x^3 - 7x + 6
=x^3 + 6 + 21 - 21 - 7x
=x^3 + 27 - 21 - 7x
=(x + 3)(x^2 - 3x + 9) - 7(3 + x)
=(x + 3)(x^2 - 3x + 9) - 7(x + 3)
=(x + 3)[(x^2 -3x + 9 - 7)
=(x + 3)(x^2 - 3x + 2)
=(x + 3)(x - 2)(x - 1)

x3 + 27

= x3 + 33

= (x + 3)(x2 – 3x + 32) [Using the identity, a3 + b3 = (a + b)(a2 – ab + b2)]

= (x + 3)(x2 – 3x + 9)



x3 – 7x + 6

= x3 – 7x + 27 – 21

= (x3 + 27) – (7x + 21)

= (x + 3)(x2 – 3x + 9) – 7(x + 3)

= (x + 3)[(x2 – 3x + 9) - 7] (Factorization)

= (x + 3)(x2 – 3x + 2)

= (x + 3)(x - 2)(x - 1)