document.write( "Question 307198: Factor the following polynomial completely.\r
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Algebra.Com's Answer #219790 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"20x%5E2%2B22xy%2B6y%5E2\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"2%2810x%5E2%2B11xy%2B3y%5E2%29\"$ Factor out the GCF $\"2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"10x%5E2%2B11xy%2B3y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"10x%5E2%2B11xy%2B3y%5E2\"$ we can see that the first term is $\"10x%5E2\"$ and the last term is $\"3y%5E2\"$ where the coefficients are 10 and 3 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 10 and the last coefficient 3 to get 30. Now what two numbers multiply to 30 and add to the middle coefficient 11? Let's list all of the factors of 30:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 30:\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-10,-15,-30 ...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 30\r
\n" ); document.write( "\n" ); document.write( "1*30\r
\n" ); document.write( "\n" ); document.write( "2*15\r
\n" ); document.write( "\n" ); document.write( "3*10\r
\n" ); document.write( "\n" ); document.write( "5*6\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-10)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-6)\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 11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 11\r
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First Number	Second Number	Sum
1	30	1+30=31
2	15	2+15=17
3	10	3+10=13
5	6	5+6=11
-1	-30	-1+(-30)=-31
-2	-15	-2+(-15)=-17
-3	-10	-3+(-10)=-13
-5	-6	-5+(-6)=-11

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 5 and 6 add up to 11 and multiply to 30\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"10x%5E2%2B11xy%2B3y%5E2\"$ , replace $\"11xy\"$ with $\"5xy%2B6xy\"$ (notice $\"5xy%2B6xy\"$ adds up to $\"11xy\"$ . So it is equivalent to $\"11xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"10x%5E2%2Bhighlight%285xy%2B6xy%29%2B3y%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"10x%5E2%2B5xy%2B6xy%2B3y%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%2810x%5E2%2B5xy%29%2B%286xy%2B3y%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"5x%282x%2By%29%2B3y%282x%2By%29\"$ Factor out the GCF of $\"5x\"$ out of the first group. Factor out the GCF of $\"3y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%285x%2B3y%29%282x%2By%29\"$ Since we have a common term of $\"2x%2By\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"10x%5E2%2B5xy%2B6xy%2B3y%5E2\"$ factors to $\"%285x%2B3y%29%282x%2By%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"10x%5E2%2B11xy%2B3y%5E2\"$ factors to $\"%285x%2B3y%29%282x%2By%29\"$ (since $\"10x%5E2%2B11xy%2B3y%5E2\"$ is equivalent to $\"10x%5E2%2B5xy%2B6xy%2B3y%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"2%2810x%5E2%2B11xy%2B3y%5E2%29\"$ and factors further to $\"2%285x%2B3y%29%282x%2By%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"20x%5E2%2B22xy%2B6y%5E2\"$ factors to $\"2%285x%2B3y%29%282x%2By%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"20x%5E2%2B22xy%2B6y%5E2=2%285x%2B3y%29%282x%2By%29\"$ \n" ); document.write( "

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