奧林匹克數學６條

1.
設y=x^2+3x+1
so 原式=(y-3)(y+3)-16
=y^2-9-16
=y^2-25
=(y+5)(y-5)
=(x^2+3x+6)(x^2+3x-4)
=(x^2+3x+6)(x-1)(x+3)

2.
(x-1)(x+2)(x-3)(x+4) + 24
=(x^2+x-2)(x^2+x-12)+24
設y=x^2+x-7
so 原式=(y-5)(y+5)+24
=y^2-25+24
=y^2-1
=(y-1)(y+1)
=(x^2+x-8)(x^2+x-6)
=(x^2+x-8)(x-2)(x+3)

3.
4(x+5)(x+6)(x+10)(x+12) - 3x^2
=4(x^2+17x+60)(x^2+16x+60)-3x^2
=(2x^2+34x+120)(2x^2+32x+120)-3x^2
設y=2x^2+33x+120
so原式=(y+x)(y-x)-3x^2
=y^2-x^2-3x^2
=y^2-4x^2
=(y-2x)(y+2x)
=(2x+31x+120)(2x+33x+120)
=(x+8)(2x+15)(2x+33x+120)

4.
(2a*2 - 3a + 1) - 22a*2+33a-23
設y=2a^2 - 3a + 1
so原式=y^2-11y-12
=(y+1)(y-12)
=(2a^2-3a+2)(2a^2-3a-11)
=(2a-1)(a-1)(2a^2-3a-11)

5.
x^3+(2y+1)x^2+(y^2+2y-1)x+y^2-1
=x^3+2x^2 y+x^2+xy^2+2xy-x+y^2-1
=(x+1)y^2+(x+1)2xy+(x+1)(x^2-1)
=(x+1)(y^2+2xy+x^2-1)
=(x+1)[(x+y)^2-1]
=(x+1)(x+y-1)(x+y+1)

6.
b^2+(a^2-a+1)b-a^3+2a^2+a-2
=b^2+a^2 b-ab+b-a^3+2a^2+a-2
=-[a^3-(b+2)a^2+(b-1)a-(b-1)(b+2]
=-[(a^2(a-b-2)+(b-1)(a-b-2)
=-(a^2+b-1)(a-b-2)