Mathematics questions

1. red balls=R, white balls=W, black balls=B
total balls=2R+1W+3B
the possible results with at least 1B
BW, WB, BR, RB, BB (order is counted)
the required probability
=(3/6)(1/5)+(1/6)(3/5)+(3/6)(2/5)+(2/6)(3/5)+(3/6)(2/5)
=2(3/6)(1/5)+2(3/6)(2/5)+(3/6)(2/5)
=1/5+2/5+1/5
=4/5

2. methods of factorization
①cross method for index of 2 (十字相乘法)
eg. 6x^2-25x+24=(3x-8)(2x-3)
steps:
first, you need to break down 6x^2 and 24 like
6x^2=(x)(6x) / (2x)(3x) (positive multiplication is enough)
24=(1)(24) / (2)(12) / (3)(8) / (4)(6) / (-1)(-24) / (-2)(-12) / (-3)(-8) / (-4)(-6)
(both positive and negative multiplication is needed)

second, you need to test for -25x
the way you test like
(I show parts of possible combinations,but you stop until required one is found)
(x)(1)+(6x)(24)=145
(x)(2)+(6x)(12)=74
(x)(3)+(6x)(8)=51
(x)(4)+(6x)(6)=40
(x)(6)+(6x)(4)=30
(x)(8)+(6x)(3)=26
(x)(12)+(6x)(2)=24
(x)(24)+(6x)(1)=30
and so on by negative multiplication and x->2x, 6x->3x
it is formal to do it, but it is so complex.
I tell you some tips or short cut.
you can see that -25x, the coefficient is -25
so positive multiplication is not needed as it always gives positive coefficient
you need to negative multiplication

third, factorization
you find (2x)(-8)+(3x)(-3)= -25
the form: (2x+a)(3x+b)
as (3x)(-3), a= -3
as (2x)(-8), b= -8
so, 6x^2-25x+24=(3x-8)(2x-3)

②testing for index 3 or more
eg. 6x^3-11x^2-x+6=(x-1)(3x+2)(2x-3)
steps:
first, you need to break down 6x^3 and 6 like
6x^3=>x,2x,3x,6x
6=1,2,3,6, -1, -2, -3, -6

second, test 6x^3-11x^2-x+6=0
when x= 1,
6(1)^3-11(1)^2-(1)+6=0
so, (x-1) is a factor
6x^3-11x^2-x+6=(x-1)(ax^2+bx+c)
(sometimes (ax^2+bx+c) cannot be factorized)
6x^3-11x^2-x+6=(x-1)(6x^2-5x-6)
(6x^2-5x-6) use cross method
6x^3-11x^2-x+6=(x-1)(3x+2)(2x-3)
or when x= 2,3,6, -1, -2, -3, -6,
or when 2x=1,2,3,6, -1, -2, -3, -6 and so on
for 6x^3-11x^2-x+6=0

as there is a word limit,
Q3 and Q4 are on comment(意見)

2013-06-22 10:27:17 補充：
3. like the description of 回答者：1 1 1 1

4. 微積分(integration and differentiation) is mostly used in physics
I = dQ/dt (I:current, Q:charge, t:time)
there are a lot of examples you search in the Internet,
it is reason why some people will say that maths serve for physics

2013-06-22 10:31:00 補充：
Q2. it is wrong you find (2x)(-8)+(3x)(-3)= -25
the correct one is you find (2x)(-8)+(3x)(-3)= -25x

3. 綫(線)性函數是一些形式為 y = ax + b 或 f(x) = ax + b 的函數，其中a和b都是常數，且a不等於0。(線性函數的定義域是實數的整體)
例如: f(x) = 2x + 1 和 y = -x + 1 都是x的線性函數。(在一個x的線性函數中，x的最高次數是1)
注意: 線性函數的圖像只會與兩條坐標軸分別相交一次，即只有一個x軸截距和一個y軸截距。