SOLUTION: Factor each difference of two squares; completely factor: x^4-16 Thanks for the help

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Question 182024This question is from textbook
: Factor each difference of two squares; completely factor:
x^4-16
Thanks for the help This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E4-16 Start with the given expression.


%28x%5E2%29%5E2-16 Rewrite x%5E4 as %28x%5E2%29%5E2.


%28x%5E2%29%5E2-%284%29%5E2 Rewrite 16 as %284%29%5E2.


Notice how we have a difference of squares A%5E2-B%5E2 where in this case A=x%5E2 and B=4.


So let's use the difference of squares formula A%5E2-B%5E2=%28A%2BB%29%28A-B%29 to factor the expression:


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


%28x%5E2%29%5E2-%284%29%5E2=%28x%5E2%2B4%29%28x%5E2-4%29 Plug in A=x%5E2 and B=4.


So this shows us that x%5E4-16 factors to %28x%5E2%2B4%29%28x%5E2-4%29.


In other words x%5E4-16=%28x%5E2%2B4%29%28x%5E2-4%29.

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Now let's factor x%5E2-4 (which is a difference of squares)


x%5E2-4 Start with the given expression.


%28x%29%5E2-4 Rewrite x%5E2 as %28x%29%5E2.


%28x%29%5E2-%282%29%5E2 Rewrite 4 as %282%29%5E2.


Notice how we have a difference of squares A%5E2-B%5E2 where in this case A=x and B=2.


So let's use the difference of squares formula A%5E2-B%5E2=%28A%2BB%29%28A-B%29 to factor the expression:


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


%28x%29%5E2-%282%29%5E2=%28x%2B2%29%28x-2%29 Plug in A=x and B=2.


So this shows us that x%5E2-4 factors to %28x%2B2%29%28x-2%29.


In other words x%5E2-4=%28x%2B2%29%28x-2%29.


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So this means that %28x%5E2%2B4%29%28x%5E2-4%29 factors even further to %28x%5E2%2B4%29%28x%2B2%29%28x-2%29



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Answer:


So x%5E4-16 completely factors to %28x%5E2%2B4%29%28x%2B2%29%28x-2%29