Question

数学因式分解题

因式分解:(x+1)4 +(x+3)4-272
请详细的讲一下,谢谢.

注:括号后的四是指四次方.

Answer 1

原式=(x+1)4-16+(x+3)4-256
=[(x+1)²+4][(x+1)²-4]+[(x+3)²+16][(x+3)²-16]
=[(x+1)²+4](x+1+2)(x+1-2)+[(x+3)²+16](x+3+4)(x+3-4)
=(x²+2x+5)(x+3)(x-1)+(x²+6x+25)(x+7)(x-1)
=(x-1)[(x³+5x²+11x+15)+(x³+13x²+67x+175)
=(x-1)(2x³+18x²+78x+190)
=(x-1)(2x³+10x²+8x²+78x+190)
=(x-1)[2x²(x+5)+2(4x+19)(x+5)]
=2(x-1)(x+5)(x²+4x+19)

Answer 2

（x+1）^4+(x+3)^4-272
=(x+3)^4-4^4+(x+1)^4-2^4
（平方差）
=[(x+3)^2-4^2][(x+3)^2+4^2]+[(x+1)^2-2^2][(x+1)^2+2^2]
（平方差）
=(x-1)(x+7)[(x+3)^2+4^2]+(x-1)(x+3)[(x+1)^2+2^2]
（提取公因式，再整理）
=2(x-1)（x^2+4x+19)(5+x)