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

An illustration of the letter 'A' in a speech bubbles

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.

Approved by eNotes Editorial Team
An illustration of the letter 'A' in a speech bubbles

`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.

Approved by eNotes Editorial Team