How can I demonstrate this is true? 1. `a^3+b^3+c^3-3abc` `=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)` 2.  `a^3+b^3+c^3` =`(a+b+c)^3-3(a+b)(b+c)(c+a)` Thank you!

Expert Answers
durbanville eNotes educator| Certified Educator

`a^3+b^3+c^3-3abc` = `(a+b+c)(a^2+b^2+c^2-ab-bc-ca)`

Expand the RHS (right-hand side):

RHS= `a^3 +ab^2+ac^2-a^2b-abc-a^2c >`

``         `+a^2b+b^3+bc^2-ab^2-b^2c-abc >`

         `+a^2c+b^2c +c^3-abc-bc^2-ac^2gt`

Rearrange and reduce. Note how the positive terms cancel out the negative terms leaving:

     `a^3+b^3+c^3 -abc-abc-abc` 

`therefore = a^3+b^3+c^3-3abc`

Ans: LHS = RHS. By expanding the right-hand side, you can show that it is equal to the left hand side.

durbanville eNotes educator| Certified Educator

In terms of eNotes rules, please note that you cannot post multiple questions. These are completely separate questions but can be answered using the same principle as explained above.

marinaart | Student

Thank you very much!I understand and i'm sorry for this.I didn't know the rules.