SOLUTION: How do you completly factor p^4 - 1, what is the answer?

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Question 150646: How do you completly factor p^4 - 1, what is the answer?
Found 3 solutions by jim_thompson5910, scott8148, vleith:
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
p%5E4+-+1 Start with the given expression.


%28p%5E2%29%5E2+-+%281%29%5E2 Rewrite p%5E4 as %28p%5E2%29%5E2. Rewrite 1 as %281%29%5E2


%28p%5E2-1%29%28p%5E2%2B1%29 Factor by use of the difference of squares.


%28p-1%29%28p%2B1%29%28p%5E2%2B1%29 Factor p%5E2-1 by use of the difference of squares.


So p%5E4+-+1 completely factors to %28p-1%29%28p%2B1%29%28p%5E2%2B1%29

Answer by scott8148(6628) About Me  (Show Source):
You can put this solution on YOUR website!
difference of squares __ (p^2+1)(p^2-1)

difference of squares (again) __ (p^2+1)(p+1)(p-1)

Answer by vleith(2983) About Me  (Show Source):
You can put this solution on YOUR website!
p%5E4+-1 is a difference of two squares.
%28p%5E2+-1%29%28p%5E2%2B1%29
Now note that p%5E2-1 is also a difference of two squares. So you can factor that the same way
%28p%5E2+-1%29%28p%5E2%2B1%29
%28p-1%29%28p%2B1%29%28p%5E2%2B1%29
It is possible to factor p%5E2%2B1 using the quadratic equation. The resulting factors contain complex roots. Unless you are at the point where you know imaginary numbers, you don't need to factor p%5E2%2B1 past where it is now.