document.write( "Question 131627: 4m^2-4m+1 \n" ); document.write( "

Algebra.Com's Answer #96074 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
Do you want to factor?\r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"4m%5E2-4m%2B1\"$ we can see that the first term is $\"4m%5E2\"$ and the last term is $\"1\"$ where the coefficients are 4 and 1 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 4 and the last coefficient 1 to get 4. Now what two numbers multiply to 4 and add to the middle coefficient -4? Let's list all of the factors of 4:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 4:\r
\n" ); document.write( "\n" ); document.write( "1,2\r
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\n" ); document.write( "\n" ); document.write( "-1,-2 ...List the negative factors as well. 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 4\r
\n" ); document.write( "\n" ); document.write( "1*4\r
\n" ); document.write( "\n" ); document.write( "2*2\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-4)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-2)\r
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\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -4? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -4\r
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First Number	Second Number	Sum
1	4	1+4=5
2	2	2+2=4
-1	-4	-1+(-4)=-5
-2	-2	-2+(-2)=-4

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -2 and -2 add up to -4 and multiply to 4\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"4m%5E2-4m%2B1\"$ , replace $\"-4m\"$ with $\"-2m%2B-2m\"$ (notice $\"-2m%2B-2m\"$ adds up to $\"-4m\"$ . So it is equivalent to $\"-4m\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"4m%5E2%2Bhighlight%28-2m%2B-2m%29%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"4m%5E2-2m-2m%2B1\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%284m%5E2-2m%29%2B%28-2m%2B1%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"2m%282m-1%29-1%282m-1%29\"$ Factor out the GCF of $\"2m\"$ out of the first group. Factor out the GCF of $\"-1\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%282m-1%29%282m-1%29\"$ Since we have a common term of $\"2m-1\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"4m%5E2-2m-2m%2B1\"$ factors to $\"%282m-1%29%282m-1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"4m%5E2-4m%2B1\"$ factors to $\"%282m-1%29%282m-1%29\"$ (since $\"4m%5E2-4m%2B1\"$ is equivalent to $\"4m%5E2-2m-2m%2B1\"$ )\r
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\n" ); document.write( "\n" ); document.write( "note: $\"%282m-1%29%282m-1%29\"$ is equivalent to $\"%282m-1%29%5E2\"$ since the term $\"2m-1\"$ occurs twice. So $\"4m%5E2-4m%2B1\"$ also factors to $\"%282m-1%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"4m%5E2-4m%2B1\"$ factors to $\"%282m-1%29%5E2\"$
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