因式分解maths

a^2+3b^2+4ab+2ac+6bc-4b+4c-4= (a^2 + 4ab + 3b^2) + (2ac + 6bc) + 4c - 4b - 4= (a + b)(a + 3b) + 2c(a + 3b) + 4c - 4b - 4= (a + 3b)(a + b + 2c) + 4c - 4b - 4= (a + 3b)(a + b + 2c) + 2[(a + b + 2c) - (a + 3b)] - 4令 x = a + 3b , y = a + b + 2c := xy + 2(y - x) - 4= xy + 2y - 2x - 4= y(x + 2) - 2(x + 2)= (y - 2) (x + 2)= (a + b + 2c - 2) (a + 3b + 2)


2010-12-01 23:19:33 補充：
另解 :

如覺拆項太抽象,可用十字乘法。

視全式為 a 的二次多項式將原式寫為 :

a^2+3b^2+4ab+2ac+6bc-4b+4c-4

= a^2 + (4ab + 2ac) + (3b^2 + 6bc - 4b + 4c - 4)

= a^2 + (4b + 2c)a + [3b^2 + (6c - 4)b + 4(c - 1)]

把它當作 x^2 + mx + n 來分解。

為此我們需先把 3b^2 + (6c - 4)b + 4(c - 1) 作因式分解 :

2010-12-01 23:19:51 補充：
用十字乘法 :

3b.......... 2
.......X
1b..........2(c - 1)
_________________________
3b * 2(c-1) + 1b * 2 = (6c - 4)b

所以 3b^2 + (6c - 4)b + 4(c - 1)
= (3b + 2)(b + 2(c - 1))
= (3b + 2)(b + 2c - 2)

原式 = a^2 + (4b + 2c)a + (3b + 2)(b + 2c - 2)

2010-12-01 23:19:56 補充：
再來一次十字乘法 :

a..........(3b + 2)
.....X
a..........(b + 2c - 2)
___________________________

a(3b+2) + a(b+2c-2) = (4b + 2c)a

故原式 = (a + 3b + 2) (a + b + 2c - 2)

What is the extent of the question??
The steps are too long,I have nver seen them.

I just wonder which teacher can give such a long chain of terms for students to factorize?

However, we can guess from the terms that we should focus on "a" and "b" using Factor Method, then see how to work on "c" and the constant is considered the least, I think.

Usually those terms with product of two variables and those square terms would be analyzed first. for they have higher chance of factorization together.
a^2 + 3b^2 + 4ab + 2ac + 6bc - 4b + 4c - 4

= ( a^2 + 4ab + 3b^2 ) + ( 2ac + 6bc ) - 4b + 4c - 4

= ( a + 3b )( a + b ) + 2c( a + 3b ) - 4b + 4c - 4

= ( a + 3b )[ ( a + b ) + 2c ] - 4b + 4c - 4

= ( a + 3b )( a + b + 2c ) + 2( a + b + 2c ) - 2a - 6b - 4

= ( a + 3b )( a + b + 2c ) + 2( a + b + 2c ) - 2( a + 3b + 2 )

= ( a + b + 2c ) [ ( a + 3b ) + 2 ] - 2( a + 3b + 2 )

= ( a + 3b + 2 )( a + b + 2c ) - 2( a + 3b + 2 )

= ( a + 3b + 2 )[ ( a + b + 2c ) - 2 ]

= ( a + 3b + 2 )( a + b + 2c - 2 )
Hope I can help you.

2010-12-06 21:11:38 補充：
Explain to the guy who may not understand if I skip a few steps.