Question 130624
Looking at {{{x^2-2x+1}}} we can see that the first term is {{{x^2}}} 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


<table border="1"><th>First Number</th><th>Second Number</th><th>Sum</th><tr><td align="center">1</td><td align="center">1</td><td>1+1=2</td></tr><tr><td align="center">-1</td><td align="center">-1</td><td>-1+(-1)=-2</td></tr></table>



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



Now looking at the expression {{{x^2-2x+1}}}, replace {{{-2x}}} with {{{-x-x}}} (notice {{{-x-x}}} adds up to {{{-2x}}}. So it is equivalent to {{{-2x}}})


{{{x^2+highlight(-x-x)+1}}}



Now let's factor {{{x^2-x-x+1}}} by grouping:



{{{(x^2-x)+(-x+1)}}} Group like terms



{{{x(x-1)-1(x-1)}}} Factor out the GCF of {{{x}}} out of the first group. Factor out the GCF of {{{-1}}} out of the second group



{{{(x-1)(x-1)}}} Since we have a common term of {{{x-1}}}, we can combine like terms


So {{{x^2-x-x+1}}} factors to {{{(x-1)(x-1)}}}



So this also means that {{{x^2-2x+1}}} factors to {{{(x-1)(x-1)}}} (since {{{x^2-2x+1}}} is equivalent to {{{x^2-x-x+1}}})



note:  {{{(x-1)(x-1)}}} is equivalent to  {{{(x-1)^2}}} since the term {{{x-1}}} occurs twice. So {{{x^2-2x+1}}} also factors to {{{(x-1)^2}}}




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     Answer:

So {{{x^2-2x+1}}} factors to {{{(x-1)^2}}}