document.write( "Question 156341: 7. Ok this is impossibly hard for me. How do I complete this using factoring? Thank you so much.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "18xy^3 + 3xy^2 - 10xy \n" ); document.write( "

Algebra.Com's Answer #115147 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
$\"18xy%5E3%2B3xy%5E2-10xy\"$ Start with the given expression\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"xy%2818y%5E2%2B3y-10%29\"$ Factor out the GCF $\"xy\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"18xy%5E2%2B3y-10\"$ \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 $\"18y%5E2%2B3y-10\"$ we can see that the first term is $\"18y%5E2\"$ and the last term is $\"-10\"$ where the coefficients are 18 and -10 respectively.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 18 and the last coefficient -10 to get -180. Now what two numbers multiply to -180 and add to the middle coefficient 3? Let's list all of the factors of -180:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of -180:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,9,10,12,15,18,20,30,36,45,60,90,180\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-9,-10,-12,-15,-18,-20,-30,-36,-45,-60,-90,-180 ...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 -180\r
\n" ); document.write( "\n" ); document.write( "(1)*(-180)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-90)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-60)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-45)\r
\n" ); document.write( "\n" ); document.write( "(5)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(9)*(-20)\r
\n" ); document.write( "\n" ); document.write( "(10)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(12)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(180)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(90)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(60)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(45)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(36)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(30)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(20)\r
\n" ); document.write( "\n" ); document.write( "(-10)*(18)\r
\n" ); document.write( "\n" ); document.write( "(-12)*(15)\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 3? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 3\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "\n" ); document.write( "

First Number	Second Number	Sum
1	-180	1+(-180)=-179
2	-90	2+(-90)=-88
3	-60	3+(-60)=-57
4	-45	4+(-45)=-41
5	-36	5+(-36)=-31
6	-30	6+(-30)=-24
9	-20	9+(-20)=-11
10	-18	10+(-18)=-8
12	-15	12+(-15)=-3
-1	180	-1+180=179
-2	90	-2+90=88
-3	60	-3+60=57
-4	45	-4+45=41
-5	36	-5+36=31
-6	30	-6+30=24
-9	20	-9+20=11
-10	18	-10+18=8
-12	15	-12+15=3

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From this list we can see that -12 and 15 add up to 3 and multiply to -180\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"18xy%5E2%2B3y-10\"$ , replace $\"3y\"$ with $\"-12y%2B15y\"$ (Remember, if $\"-12%2B15=3\"$ , then $\"-12y%2B15y=3y\"$ )\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"18y%5E2%2Bhighlight%28-12y%2B15y%29%2B-10\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's factor $\"18y%5E2-12y%2B15y-10\"$ by grouping:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%2818y%5E2-12y%29%2B%2815y-10%29\"$ Group like terms\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"6y%283y-2%29%2B5%283y-2%29\"$ Factor out the GCF of $\"6y\"$ out of the first group. Factor out the GCF of $\"5\"$ out of the second group\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So this also means that $\"18y%5E2%2B3y-10\"$ factors to $\"%286y%2B5%29%283y-2%29\"$ (since $\"18y%5E2%2B3y-10\"$ is equivalent to $\"18y%5E2-12y%2B15y-10\"$ )\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( "So the expression $\"xy%2818y%5E2%2B3y-10%29\"$ factors further to $\"xy%286y%2B5%29%283y-2%29\"$ \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( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"18xy%5E3%2B3xy%5E2-10xy\"$ completely factors to $\"xy%286y%2B5%29%283y-2%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To check the answer, simply FOIL and expand $\"xy%286y%2B5%29%283y-2%29\"$ to get $\"18xy%5E3%2B3xy%5E2-10xy\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );