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Question

(a-b)(a-b+c)      =      (b-a)(a-b+c)

(b-a)(c-b)(d-c)        (a-b)(b-c)(c-d)

Can this be proven?  

My opinion is that the part in bold is incorrect.  

Please verify.

Thank-you!

Answer 1

It is quite simple.

Use the following fact

x - y = -1 * (y - x)

The result is as follows

(a-b)(a-b+c)                  -1*(b-a)(a-b+c)

-----------------  =  ------------------------------------

(b-a)(c-b)(d-c)       [-1*(a-b)][-1*(b-c)][-1*(c-d)]

 

    -1 * (b-a)(a-b+c)               (b-a)(a-b+c)

= --------------------------  =  ------------------

    (-1)³ * (a-b)(b-c)(c-d)       (a-b)(b-c)(c-d)

 

The last step works because -1 / (-1)³ = 1.