SOLUTION: Factorise the following (polynomial): (a+b)^3-(a-b)^3

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Question 864944: Factorise the following (polynomial):
(a+b)^3-(a-b)^3

Found 2 solutions by ben720, Alan3354:
Answer by ben720(159) About Me  (Show Source):
You can put this solution on YOUR website!
First, lengthen the exponent:
%28a%2Bb%29%28a%2Bb%29%28a%2Bb%29-%28a-b%29%28a-b%29%28a-b%29
To find (a+b)(a+b)(a+b), find all the multiplications and add them:
a*a*a
a*a*b
a*b*a
a*b*b
b*a*a
b*a*b
b*b*a
b*b*b
for a total of a%5E3%2Ba%5E2b%2Ba%5E2b%2Bab%5E2%2Ba%5E2b%2Bab%5E2%2Bab%5E2%2Bb%5E3, or a%5E3%2B3a%5E2b%2B3ab%5E2%2Bb%5E3
For (a-b)(a-b)(a-b), it's the same, except the terms with exactly 1 or 3 b's will be negative.
The total of (a-b)(a-b)(a-b) is a%5E3-3a%5E2b%2B3ab%5E2-b%5E3

Because you're subtracting the second, your number is
%28a%5E3%2B3a%5E2b%2B3ab%5E2%2Bb%5E3%29-%28a%5E3-3a%5E2b%2B3ab%5E2-b%5E3%29
Distribute the -1
a%5E3%2B3a%5E2b%2B3ab%5E2%2Bb%5E3-a%5E3%2B3a%5E2b-3ab%5E2%2Bb%5E3
Combine like terms
6a%5E2b%2B2b%5E3

Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
Factorise the following (polynomial):
(a+b)^3-(a-b)^3
===============
It's a difference of 2 cubes.
m%5E3+-+n%5E3+=+%28m-n%29%2A%28m%5E2+%2B+mn+%2B+n%5E2%29
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(a+b)^3-(a-b)^3
=
= 2b%2A%28a%5E2+%2B+2ab+%2B+b%5E2+%2B+a%5E2+-+b%5E2+%2B+a%5E2+-+2ab+%2B+b%5E2%29
= 2b%2A%283a%5E2+%2B+b%5E2%29
= 6a%5E2b+%2B+2b%5E3