document.write( "Question 407015: Please help, \r
\n" ); document.write( "\n" ); document.write( "factor the expression 81a^2-36ab+4b^2 into a product of binomials.
\n" ); document.write( " \n" ); document.write( "

Algebra.Com's Answer #287243 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "Looking at $\"81a%5E2-36ab%2B4b%5E2\"$ we can see that the first term is $\"81a%5E2\"$ and the last term is $\"4b%5E2\"$ where the coefficients are 81 and 4 respectively.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 81 and the last coefficient 4 to get 324. Now what two numbers multiply to 324 and add to the middle coefficient -36? Let's list all of the factors of 324:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of 324:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,9,12,18,27,36,54,81,108,162\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-9,-12,-18,-27,-36,-54,-81,-108,-162 ...List the negative factors as well. This will allow us to find all possible combinations\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to 324\r
\n" ); document.write( "\n" ); document.write( "1*324\r
\n" ); document.write( "\n" ); document.write( "2*162\r
\n" ); document.write( "\n" ); document.write( "3*108\r
\n" ); document.write( "\n" ); document.write( "4*81\r
\n" ); document.write( "\n" ); document.write( "6*54\r
\n" ); document.write( "\n" ); document.write( "9*36\r
\n" ); document.write( "\n" ); document.write( "12*27\r
\n" ); document.write( "\n" ); document.write( "18*18\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-324)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-162)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-108)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-81)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-54)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(-12)*(-27)\r
\n" ); document.write( "\n" ); document.write( "(-18)*(-18)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -36? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -36\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	324	1+324=325
2	162	2+162=164
3	108	3+108=111
4	81	4+81=85
6	54	6+54=60
9	36	9+36=45
12	27	12+27=39
18	18	18+18=36
-1	-324	-1+(-324)=-325
-2	-162	-2+(-162)=-164
-3	-108	-3+(-108)=-111
-4	-81	-4+(-81)=-85
-6	-54	-6+(-54)=-60
-9	-36	-9+(-36)=-45
-12	-27	-12+(-27)=-39
-18	-18	-18+(-18)=-36

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From this list we can see that -18 and -18 add up to -36 and multiply to 324\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"81a%5E2-36ab%2B4b%5E2\"$ , replace $\"-36ab\"$ with $\"-18ab%2B-18ab\"$ (notice $\"-18ab%2B-18ab\"$ adds up to $\"-36ab\"$ . So it is equivalent to $\"-36ab\"$ )\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"81a%5E2%2Bhighlight%28-18ab%2B-18ab%29%2B4b%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"81a%5E2-18ab-18ab%2B4b%5E2\"$ by grouping:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%2881a%5E2-18ab%29%2B%28-18ab%2B4b%5E2%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"9a%289a-2b%29-2b%289a-2b%29\"$ Factor out the GCF of $\"9a\"$ out of the first group. Factor out the GCF of $\"-2b\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%289a-2b%29%289a-2b%29\"$ Since we have a common term of $\"9a-2b\"$ , we can combine like terms\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"81a%5E2-18ab-18ab%2B4b%5E2\"$ factors to $\"%289a-2b%29%289a-2b%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this also means that $\"81a%5E2-36ab%2B4b%5E2\"$ factors to $\"%289a-2b%29%289a-2b%29\"$ (since $\"81a%5E2-36ab%2B4b%5E2\"$ is equivalent to $\"81a%5E2-18ab-18ab%2B4b%5E2\"$ )\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note: $\"%289a-2b%29%289a-2b%29\"$ is equivalent to $\"%289a-2b%29%5E2\"$ since the term $\"9a-2b\"$ occurs twice. So $\"81a%5E2-36ab%2B4b%5E2\"$ also factors to $\"%289a-2b%29%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "------------------------------------------------------------\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"81a%5E2-36ab%2B4b%5E2\"$ factors to $\"%289a-2b%29%5E2\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
\n" ); document.write( "
\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
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

\n" );