SOLUTION: how to factor 16x^4-1

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Question 214414: how to factor 16x^4-1
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
16x%5E4-1 Start with the given expression.


%284x%5E2%29%5E2-1 Rewrite 16x%5E4 as %284x%5E2%29%5E2.


%284x%5E2%29%5E2-%281%29%5E2 Rewrite 1 as %281%29%5E2


Take note that we have a difference of squares A%5E2-B%5E2 in which A=4x%5E2 and B=1. We can factor this difference of squares using the formula A%5E2-B%5E2=%28A-B%29%28A%2BB%29


A%5E2-B%5E2=%28A-B%29%28A%2BB%29 Start with the difference of squares formula


%284x%5E2%29%5E2-%281%29%5E2=%284x%5E2-1%29%284x%5E2%2B1%29 Plug in A=4x%5E2 and B=1


So 16x%5E4-1 factors to %284x%5E2-1%29%284x%5E2%2B1%29


We can keep going and factor 4x%5E2-1 further.


4x%5E2-1


%282x%29%5E2-%281%29%5E2 Rewrite 4x%5E2 as %282x%29%5E2. Rewrite 1 as %281%29%5E2

Notice that we again have a difference of squares A%5E2-B%5E2 in which A=2x and B=1. We can factor this difference of squares using the formula A%5E2-B%5E2=%28A-B%29%28A%2BB%29


A%5E2-B%5E2=%28A-B%29%28A%2BB%29 Start with the difference of squares formula


%282x%29%5E2-%281%29%5E2=%282x-1%29%282x%2B1%29 Plug in A=2x and B=1


So 4x%5E2-1 factors to %282x-1%29%282x%2B1%29


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Answer:

Putting all of this together tells us that 16x%5E4-1 completely factors to %282x-1%29%282x%2B1%29%284x%5E2%2B1%29