document.write( "Question 387326: Can someone tell me how to factor polynomials by grouping, such as:\r
\n" ); document.write( "\n" ); document.write( " $\"36a%5E2-36ab-27b%5E2\"$ \r
\n" ); document.write( "\n" ); document.write( "This is part of my take home final and is really stressing me out! \n" ); document.write( "

Algebra.Com's Answer #273789 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( " $\"36a%5E2-36ab-27b%5E2\"$ Start with the given expression.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"9%284a%5E2-4ab-3b%5E2%29\"$ Factor out the GCF $\"9\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"4a%5E2-4ab-3b%5E2\"$ \r
\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( "Looking at the expression $\"4a%5E2-4ab-3b%5E2\"$ , we can see that the first coefficient is $\"4\"$ , the second coefficient is $\"-4\"$ , and the last coefficient is $\"-3\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"4\"$ by the last coefficient $\"-3\"$ to get $\"%284%29%28-3%29=-12\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-12\"$ (the previous product) and add to the second coefficient $\"-4\"$ ?\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"-12\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"-12\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,12\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-12\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. This will allow us to find all possible combinations.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to $\"-12\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-12) = -12
\n" ); document.write( "2*(-6) = -12
\n" ); document.write( "3*(-4) = -12
\n" ); document.write( "(-1)*(12) = -12
\n" ); document.write( "(-2)*(6) = -12
\n" ); document.write( "(-3)*(4) = -12\r
\n" ); document.write( "
\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 $\"-4\"$ :\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	-12	1+(-12)=-11
2	-6	2+(-6)=-4
3	-4	3+(-4)=-1
-1	12	-1+12=11
-2	6	-2+6=4
-3	4	-3+4=1

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"2\"$ and $\"-6\"$ add to $\"-4\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"2\"$ and $\"-6\"$ both multiply to $\"-12\"$ and add to $\"-4\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-4ab\"$ with $\"2ab-6ab\"$ . Remember, $\"2\"$ and $\"-6\"$ add to $\"-4\"$ . So this shows us that $\"2ab-6ab=-4ab\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"4a%5E2%2Bhighlight%282ab-6ab%29-3b%5E2\"$ Replace the second term $\"-4ab\"$ with $\"2ab-6ab\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%284a%5E2%2B2ab%29%2B%28-6ab-3b%5E2%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2a%282a%2Bb%29%2B%28-6ab-3b%5E2%29\"$ Factor out the GCF $\"2a\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"2a%282a%2Bb%29-3b%282a%2Bb%29\"$ Factor out $\"-3b\"$ from the second group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%282a-3b%29%282a%2Bb%29\"$ Factor $\"2a%2Bb\"$ from the entire expression.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"4a%5E2-4ab-3b%5E2\"$ factors to $\"%282a-3b%29%282a%2Bb%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "This means that $\"36a%5E2-36ab-27b%5E2\"$ completely factors to $\"9%282a-3b%29%282a%2Bb%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"36a%5E2-36ab-27b%5E2=9%282a-3b%29%282a%2Bb%29\"$ \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, feel free to check out my tutoring website\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

\n" );