Question

4(x+5)(x+6)(x+10)(x+12)-3x^2 这个式子怎么分解因式

Answer 1

f(x)=4(x+5)(x+6)(x+10)(x+12)-3x^2
可以看出x=-8是f(x)的一个根，所以因式包含x+8
令a=x+8
f(x)=4(a-3)(a-2)(a+2)(a+4)-3(a-8)^2
=4[(a-3)(a+4)][(a-2)(a+2)]-3(a-8)^2
=4(a^2+a-12)(a^2-4)-3(a-8)^2
=4a^4+4a^3-67a^2+32a
=a(4a^3+4a^2-67a+32)
=a(4a^3-2a^2+6a^2-67a+32)
=a(2(a^2)(2a-1)+(2a-1)(3a-32))
=a(2a-1)(2a^2+3a-32)

到此己不能在有理数域分解。代回a=x+8
f(x)=(x+8)(2x+15)(2x^2+35x+120)

Answer 2

原式=4[(x+5)(x+12)][(x+6)(x+10)]-3x²
=4[(x²+60)+17x][(x²+60)+16x]-3x²
=4(x²+60)²+132x²(x²+60)+1088x²-3x²
=4(x²+60)²+132x²(x²+60)+1085x²
=[2(x²+60)+31][2(x²+60)+35]
=(2x²+120x+31)(2x²+120x+35)