SOLUTION: Just a homework question I've spent an hour on already without any clue how to answer. I tried the foil but I must be doing something wrong. (a-b)(a^2+ab+b^2

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Just a homework question I've spent an hour on already without any clue how to answer. I tried the foil but I must be doing something wrong. (a-b)(a^2+ab+b^2      Log On


   

Question 881206: Just a homework question I've spent an hour on already without any clue how to answer. I tried the foil but I must be doing something wrong.
(a-b)(a^2+ab+b^2)
Found 2 solutions by jim_thompson5910, KMST:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29


a%28a%5E2%2Bab%2Bb%5E2%29-b%28a%5E2%2Bab%2Bb%5E2%29 Distribute (use the idea that %28a%2Bb%29%28c%2Bd%29=a%28c%2Bd%29%2Bb%28c%2Bd%29


a%5E3%2Ba%5E2b%2Bab%5E2-a%5E2b-ab%5E2-b%5E3 Distribute


a%5E3-b%5E3 Combine like terms.


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Special Note:


This is the difference of cubes factoring formula


a%5E3-b%5E3=%28a-b%29%28a%5E2%2Bab%2Bb%5E2%29


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So %28a-b%29%28a%5E2%2Bab%2Bb%5E2%29 simplifies to a%5E3-b%5E3

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
When you multiply factors with 2 or more terms, you end up making all the possible combinations of one term from one factor and another term from the other factor.
It is like picking shorts and shirts for an outfit.
(Black+Gray+Red) shorts, times (yellow+white+pink) shirts make
(B+G+R)(y+w+p)=By+Bw+Bp+Gy+Gw+Gp+Ry+Rw+Rp

You can only FOIL when each factor has 2 terms, because then you can make combinations so we need a 6-letter acronym, like FOIL@#.
No one has thought of cute names to call the products of the middle term of the 3-term factor (in this case ab ) times the first ( a ) and last ( b ) terms of a 2-term factor.
No one has thought of letter names for the 9 products formed when both factors have 3 terms.
Maybe it is lucky that we do not have to memorize more recipes to be applied without thinking or understanding.