SOLUTION: Factor the trinomial {{{x^2 - 2x + 1}}}.

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Question 130624: Factor the trinomial x%5E2+-+2x+%2B+1.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
Looking at x%5E2-2x%2B1 we can see that the first term is x%5E2 and the last term is 1 where the coefficients are 1 and 1 respectively.

Now multiply the first coefficient 1 and the last coefficient 1 to get 1. Now what two numbers multiply to 1 and add to the middle coefficient -2? Let's list all of the factors of 1:



Factors of 1:
1

-1 ...List the negative factors as well. This will allow us to find all possible combinations

These factors pair up and multiply to 1
1*1
(-1)*(-1)

note: remember two negative numbers multiplied together make a positive number


Now which of these pairs add to -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2

First NumberSecond NumberSum
111+1=2
-1-1-1+(-1)=-2



From this list we can see that -1 and -1 add up to -2 and multiply to 1


Now looking at the expression x%5E2-2x%2B1, replace -2x with -x-x (notice -x-x adds up to -2x. So it is equivalent to -2x)

x%5E2%2Bhighlight%28-x-x%29%2B1


Now let's factor x%5E2-x-x%2B1 by grouping:


%28x%5E2-x%29%2B%28-x%2B1%29 Group like terms


x%28x-1%29-1%28x-1%29 Factor out the GCF of x out of the first group. Factor out the GCF of -1 out of the second group


%28x-1%29%28x-1%29 Since we have a common term of x-1, we can combine like terms

So x%5E2-x-x%2B1 factors to %28x-1%29%28x-1%29


So this also means that x%5E2-2x%2B1 factors to %28x-1%29%28x-1%29 (since x%5E2-2x%2B1 is equivalent to x%5E2-x-x%2B1)


note: %28x-1%29%28x-1%29 is equivalent to %28x-1%29%5E2 since the term x-1 occurs twice. So x%5E2-2x%2B1 also factors to %28x-1%29%5E2



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Answer:
So x%5E2-2x%2B1 factors to %28x-1%29%5E2