SOLUTION: What is the answer to (a+1)(a+1)(a-1)(a-1)

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Question 400343: What is the answer to
(a+1)(a+1)(a-1)(a-1)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
(a+1)(a+1)(a-1)(a-1)
The straightforward way to do this is to:
  1. Multiply the first two factors, (a+1)(a+1)
  2. Multiply the answer from step #1 by the next factor: (a-1)
  3. Multiply the answer from step #2 by the next factor: (a-1)

But there is a faster way. Since this is multiplication we can use both the Commutative and Associative Properties to rearrange the order and grouping. If we rewrite the expression as:
((a+1)(a-1))((a+1)(a-1))
we have two identical multiplications of (a+1)(a-1) which should save some time. And (a+1)(a-1) is easy to multiply using the %28x%2By%29%28x-y%29+=+x%5E2+-+y%5E2 pattern. (You could use FOIL instead of the pattern.) So rewriting the expression this way makes what we are going to do both faster and easier! Using the pattern (twice) on (a+1)(a-1) we get:
%28a%5E2+-+1%5E2%29%28a%5E2+-+1%5E2%29
which simplifies to:
%28a%5E2+-+1%29%28a%5E2+-+1%29
Now we have a multiplication of two identical binomials. For this we can use FOIL or, better yet, use the %28x-y%29%28x-y%29+=+x%5E2+-2xy+%2B+y%5E2 pattern:
%28a%5E2%29%5E2+-+2%28a%5E2%29%281%29+%2B+%281%29%5E2
which simplifies to:
a%5E4+-+2a%5E2+%2B+1
This is the simplified answer regardless of whether you
  • did the problem the straightforward way or not
  • Used patterns or FOIL to multiply.

But the way I did it here is probably the fastest and easiest (if you know your patterns).