document.write( "Question 631477: completely factor the expression r^2-2r+1
\n" ); document.write( "a. prime
\n" ); document.write( "b. r(r-2)+1
\n" ); document.write( "c. (r-1)(r-1)
\n" ); document.write( "d. (r+1)(r-1)\r
\n" ); document.write( "\n" ); document.write( "If you could please explain I would really appreciate it \n" ); document.write( "

Algebra.Com's Answer #397597 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( "Looking at the expression $\"r%5E2-2r%2B1\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-2\"$ , and the last term is $\"1\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"1\"$ to get $\"%281%29%281%29=1\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"1\"$ (the previous product) and add to the second coefficient $\"-2\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"1\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"1\"$ :\r
\n" ); document.write( "\n" ); document.write( "1\r
\n" ); document.write( "\n" ); document.write( "-1\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"1\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*1 = 1
\n" ); document.write( "(-1)*(-1) = 1\r
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\n" ); document.write( "\n" ); document.write( "Now let's add up each pair of factors to see if one pair adds to the middle coefficient $\"-2\"$ :\r
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First Number	Second Number	Sum
1	1	1+1=2
-1	-1	-1+(-1)=-2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-1\"$ and $\"-1\"$ add to $\"-2\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-1\"$ and $\"-1\"$ both multiply to $\"1\"$ and add to $\"-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-2r\"$ with $\"-r-r\"$ . Remember, $\"-1\"$ and $\"-1\"$ add to $\"-2\"$ . So this shows us that $\"-r-r=-2r\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"r%5E2%2Bhighlight%28-r-r%29%2B1\"$ Replace the second term $\"-2r\"$ with $\"-r-r\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28r%5E2-r%29%2B%28-r%2B1%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r-1%29%2B%28-r%2B1%29\"$ Factor out the GCF $\"r\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r-1%29-1%28r-1%29\"$ Factor out $\"1\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r
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\n" ); document.write( "\n" ); document.write( " $\"%28r-1%29%28r-1%29\"$ Combine like terms. Or factor out the common term $\"r-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"r%5E2-2r%2B1\"$ factors to $\"%28r-1%29%28r-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"r%5E2-2r%2B1=%28r-1%29%28r-1%29\"$ for all values of 'r'\r
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\n" ); document.write( "\n" ); document.write( "So the answer is choice c).\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28r-1%29%5E2\"$ to get $\"r%5E2-2r%2B1\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
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\n" ); document.write( "\n" ); document.write( "Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you\r
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\n" ); document.write( "\n" ); document.write( "Jim
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