document.write( "Question 407310: Can someone please help?\r
\n" ); document.write( "\n" ); document.write( "factor the expression 81x^2-54xy+9y^2 into a product of binomials. \n" ); document.write( "

Algebra.Com's Answer #287194 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"81x%5E2-54xy%2B9y%5E2\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"9%289x%5E2-6xy%2By%5E2%29\"$ Factor out the GCF $\"9\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"9x%5E2-6xy%2By%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"9x%5E2-6xy%2By%5E2\"$ we can see that the first term is $\"9x%5E2\"$ and the last term is $\"y%5E2\"$ where the coefficients are 9 and 1 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 9 and the last coefficient 1 to get 9. Now what two numbers multiply to 9 and add to the middle coefficient -6? Let's list all of the factors of 9:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 9:\r
\n" ); document.write( "\n" ); document.write( "1,3\r
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\n" ); document.write( "\n" ); document.write( "-1,-3 ...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 9\r
\n" ); document.write( "\n" ); document.write( "1*9\r
\n" ); document.write( "\n" ); document.write( "3*3\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-9)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-3)\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 -6? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -6\r
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First Number	Second Number	Sum
1	9	1+9=10
3	3	3+3=6
-1	-9	-1+(-9)=-10
-3	-3	-3+(-3)=-6

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -3 and -3 add up to -6 and multiply to 9\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"9x%5E2-6xy%2By%5E2\"$ , replace $\"-6xy\"$ with $\"-3xy-3xy\"$ (notice $\"-3xy-3xy\"$ combines back to $\"-6xy\"$ . So it is equivalent to $\"-6xy\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"9x%5E2%2Bhighlight%28-3xy-3xy%29%2By%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"9x%5E2-3xy-3xy%2By%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%289x%5E2-3xy%29%2B%28-3xy%2By%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%283x-y%29-y%283x-y%29\"$ Factor out the GCF of $\"3x\"$ out of the first group. Factor out the GCF of $\"-y\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x-y%29%283x-y%29\"$ Since we have a common term of $\"3x-y\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"9x%5E2-3xy-3xy%2By%5E2\"$ factors to $\"%283x-y%29%283x-y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"9x%5E2-6xy%2By%5E2\"$ factors to $\"%283x-y%29%283x-y%29\"$ (since $\"9x%5E2-6xy%2By%5E2\"$ is equivalent to $\"9x%5E2-3xy-3xy%2By%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( "note: $\"%283x-y%29%283x-y%29\"$ is equivalent to $\"%283x-y%29%5E2\"$ since the term $\"3x-y\"$ occurs twice. So $\"9x%5E2-6xy%2By%5E2\"$ also factors to $\"%283x-y%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"9%289x%5E2-6xy%2By%5E2%29\"$ and factors further to $\"9%283x-y%29%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"81x%5E2-54xy%2B9y%5E2\"$ factors to $\"9%283x-y%29%5E2\"$ \r
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\n" ); document.write( "\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 \n" ); document.write( "

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