國高中銜接數學利用公式因數分解

答案僅供參考 未必完全正確




將下列兩數因數分解:
(1)9的6次方減64
(2)2的24次方減1

(1)運用平方差(a+b)(a-b)=a^2-b^2
9^6-64
=(9^3)^2-8^2
=729^2-8^2
=(729+8)(729-8)
=737*721


(2)運用平方差(a+b)(a-b)=a^2-b^2
2^24-1
=(2^12)^2-1^2
=(2^12+1)(2^12-1)
=(4096+1)(4096-1)
=4097*4095

用平方差公式a^2-b^2=(a+b)(a-b)
(1)9^6-2^6
=(9^3)^2-(2^3)^2
=(9^3-2^3)(9^3+2^3)
=(729-8)(729+8)
=721*737
=531377
或用立方和公式a^3+b^3=(a+b)(a^2+ab+b^2)
與立方差公式a^3-b^3=(a-b)(a^2-ab+b^2)帶入也可得出
(2)2^24-1
=(2^12)^2-1^2
=(2^12-1)(2^12+1)
=(2^6-1)(2^6+1)(2^12+1)
=(2^3-1)(2^3+1)(2^6+1)(2^12+1)
=(8-1)(8+1)(32+1)[(2^4)^3+1^3]
=7*9*33*(2^4+1)(2^8-2^4+1)
=7*9*3*11*17*113
=3993759

a² - b² = (a - b)(a + b)

a³ ± b³ = (a ± b)(a² ∓ ab + b²)