Question

数学问题，因式分解。。大家帮帮忙！！(要的是详细过程）

1.x^3-11^2+31x-21
2.4x^4-13x^2+9
3.3x^2+5xy-2y^2+x+9y-4
答案分别是：1.(x-1)(x-3)(x-7) 2.(2x+3)(2x-3)(x+1)(x-1) 3.(3x-y+4)(x+2y-1)
是11x^2

Answer 1

.x^3-11^2+31x-21
=.x^3-7x^2-4x^2+31x-21=x^2(x-7)-(4x-3)(x-7)=(x^2-4x+3)(x-7)=(x-1)(x-3)(x-7)
4x^4-13x^2+9=(4x^2-9)(x^2-1)=(2x+3)(2x-3)(x+1)(x-1)
3x^2+5xy-2y^2+x+9y-4
=(3x-y)(x+2y)+x+9y-4
=(3x-y)(x+2y)+4(x+2y)-(3x-y)-4(注意用配方法）
=(3x-y+4)(x+2y-1)

追问

4x^4-13x^2+9=(4x^2-9)(x^2-1)怎么到这步的，有公式吗？

追答

用十字相乘法，把x^2看到x,原式变为4x^2-13x+9

Answer 2

1.x^3-11x^2+31x-21
=x^3-7x^2-4x^2+28x+3x-21
=x^2(x-7)-4x(x-7)+3(x-7)
=(x-7）（x^2-4x+3)
=(x-7)(x-1)(x-3)
2.4x^4-13x^2+9
=(4x^2-9)(x^2-1)
=(2x+3)(2x-3)(x+1)(x-1)
3.3x^2+5xy-2y^2+x+9y-4
=3x^2-xy+6xy-y^2-3x+y+4x+8y-4
=x(3x-y)+2y(3x-y)-(3x-y)+4x+8y-4
=(3x-y)(x+2y-1)+4(x+2y-1)
=(3x-y+4)(x+2y-1)

Answer 3

(1)x^3-11x^2+31x-21（这个11x^2打成了11^2）
=(x^3-3x^2)+(-8x^2+24x)+(7x-21)
=x^2(x-3)-8x(x-3)+7(x-3)
=(x-3)(x^2-8x+7)
=(x-3)(x-1)(x-7)
(2)4x^4-13x^2+9
=4x^4-12x^2+9-x^2
=(2x^2-3)^2-x^2
=(2x^2+x-3)(2x^2-x-3)
=(2x+3)(x-1)*(2x-3)(x+1)
=(2x+3)(2x-3)(x+1)(x-1)
当然也可以直接分解：4x^4-13x^2+9=(4x^2-9)(x^2-1)=(2x+3)(2x-3)(x+1)(x-1)
(3)3x^2+5xy-2y^2+x+9y-4
=(3x-y)(x+2y)+x+9y-4
此时可以用待定系数法，设原式=(3x-y+a)(x+2y+b)
则原式=(3x-y)(x+2y)+a(x+2y)+b(3x-y)+ab
=(3x-y)(x+2y)+ax+3bx+2ay-by+ab
=(3x-y)(x+2y)+(a+3b)x+(2a-b)y+ab
对照原式系数，可以得到方程组a+3b=1,2a-b=9，解得a=4,b=-1
所以原式=(3x-y+4)(x+2y-1)