document.write( "Question 526716: I am lost with this question. Factor completely Remember to look first for a common factor. Check by multiplying. If a polynomial is prime, state this.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "X^2 − 11 xy + 24y^2 \n" ); document.write( "

Algebra.Com's Answer #348658 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2-11xy%2B24y%5E2\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-11\"$ , and the last coefficient is $\"24\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last coefficient $\"24\"$ to get $\"%281%29%2824%29=24\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"24\"$ (the previous product) and add to the second coefficient $\"-11\"$ ?\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 $\"24\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"24\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,24\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-24\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 $\"24\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*24 = 24
\n" ); document.write( "2*12 = 24
\n" ); document.write( "3*8 = 24
\n" ); document.write( "4*6 = 24
\n" ); document.write( "(-1)*(-24) = 24
\n" ); document.write( "(-2)*(-12) = 24
\n" ); document.write( "(-3)*(-8) = 24
\n" ); document.write( "(-4)*(-6) = 24\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 $\"-11\"$ :\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	24	1+24=25
2	12	2+12=14
3	8	3+8=11
4	6	4+6=10
-1	-24	-1+(-24)=-25
-2	-12	-2+(-12)=-14
-3	-8	-3+(-8)=-11
-4	-6	-4+(-6)=-10

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-3\"$ and $\"-8\"$ add to $\"-11\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-3\"$ and $\"-8\"$ both multiply to $\"24\"$ and add to $\"-11\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-11xy\"$ with $\"-3xy-8xy\"$ . Remember, $\"-3\"$ and $\"-8\"$ add to $\"-11\"$ . So this shows us that $\"-3xy-8xy=-11xy\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%28-3xy-8xy%29%2B24y%5E2\"$ Replace the second term $\"-11xy\"$ with $\"-3xy-8xy\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2-3xy%29%2B%28-8xy%2B24y%5E2%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x-3y%29%2B%28-8xy%2B24y%5E2%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%28x-3y%29-8y%28x-3y%29%29\"$ Factor $\"-8y\"$ from the second group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28x-8y%29%28x-3y%29%29\"$ Factor $\"x-3y\"$ from the entire expression.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"x%5E2-11xy%2B24y%5E2\"$ completely factors to $\"%28x-8y%29%28x-3y%29%29\"$ \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, 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" );