Question

关于因式分解的数学题

1.x^3+6x^2+11x+6
2.x^4+2x^3-9x^2+8
3.(y+1)^4+(y+3)^4-272
4.(xy-1)^2+(x+y-2)(x+y-2xy)
5.x^4-2x^3-27x^2-44x+7

Answer 1

1.x^3+6x^2+11x+6
=x^3+x^2+5x^2+11x+6
=x^2(x+1)+(5x+6)(x+1)
=(x^2+5x+6)(x+1)
=(x+1)(x+2)(x+3)
2.x^4+2x^3-9x^2+8=解不出
设原式=(x²+ax+2)(x²+bx+4)或=(x²+ax-2)(x²+bx-4)
=x^4+(a+b)x^3+(6+ab)x^2+(4a+2b)x+8
或=x^4+(a+b)x^3+(-6+ab)x^2-(4a+2b)x+8
a+b=2 6+ab=-9 4a+2b=0
或a+b=2 -6+ab=-9 -(4a+2b)=0
两组方程无解
不能分解因式。
3.(y+1)^4+(y+3)^4-272
令x=y+2
原式=(x-1)4+(x+1)4-272
2(x4+6x2+1)-272
=2(x4+6x2-135)
=2(x2-9)(x2+15)
=2(x+3)(x-3)(x2+15)
=2(y+5)(y-1)(y2+4y+19)
4.(xy-1)^2+(x+y-2)(x+y-2xy)
=(x+y)²-2(1+xy)(x+y)+4xy+1-2xy+x²y²
=(x+y)²-2(1+xy)(x+y)+(1+xy)²
=(x+y)²-(1+xy)(x+y)-(1+xy)(x+y)+(1+xy)²
=(x+y)(x+y-1-xy)-(1+xy)(x+y-1-xy)
=(x+y-1-xy)(x+y-1-xy)
=(x+y-1-xy)²
5.x^4-2x^3-27x^2-44x+7
设原式=(x²+ax+1)(x²+bx+7)或=(x²+ax-1)(x²+bx-7)
x⁴+(a+b)x³+(ab+8)x²+(7a+b)x+7=x⁴-2x³-27x²-44x+7
或x⁴+(a+b)x³+(ab-8)x²-(7a+b)x+7=x⁴-2x³-27x²-44x+7
比较等式两边系数可得
a+b=-2
ab+8=-27
7a+b=-44
或a+b=-2
ab-8=-27
-7a-b=-44
解这两个方程组可得:
a=-7 b=5 [第二个方程组是无解的]
所以原式=(x²-7x+1)(x²+5x+7)

追问

第二个能解的，答案是(x+1)(x-1)(x-2)(x+4)

追答

不对，可能是你题目给错了。
(x+1)(x-1)(x-2)(x+4)
=(x^2-1)(x^2+2x-8)
=x^4+2x^3-8x^2-x^2-2x+8
=x^4+2x^3-9x^2-2x+8
与原题差-2x这一项