document.write( "Question 253128: Factoring help.\r
\n" ); document.write( "\n" ); document.write( "30v^2+186v-432 \n" ); document.write( "

Algebra.Com's Answer #185353 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( " $\"30v%5E2%2B186v-432\"$ Start with the given expression.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"6%285v%5E2%2B31v-72%29\"$ Factor out the GCF $\"6\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"5v%5E2%2B31v-72\"$ \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 $\"5v%5E2%2B31v-72\"$ , we can see that the first coefficient is $\"5\"$ , the second coefficient is $\"31\"$ , and the last term is $\"-72\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"5\"$ by the last term $\"-72\"$ to get $\"%285%29%28-72%29=-360\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-360\"$ (the previous product) and add to the second coefficient $\"31\"$ ?\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 $\"-360\"$ (the previous product).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Factors of $\"-360\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180,360\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-5,-6,-8,-9,-10,-12,-15,-18,-20,-24,-30,-36,-40,-45,-60,-72,-90,-120,-180,-360\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 $\"-360\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-360) = -360
\n" ); document.write( "2*(-180) = -360
\n" ); document.write( "3*(-120) = -360
\n" ); document.write( "4*(-90) = -360
\n" ); document.write( "5*(-72) = -360
\n" ); document.write( "6*(-60) = -360
\n" ); document.write( "8*(-45) = -360
\n" ); document.write( "9*(-40) = -360
\n" ); document.write( "10*(-36) = -360
\n" ); document.write( "12*(-30) = -360
\n" ); document.write( "15*(-24) = -360
\n" ); document.write( "18*(-20) = -360
\n" ); document.write( "(-1)*(360) = -360
\n" ); document.write( "(-2)*(180) = -360
\n" ); document.write( "(-3)*(120) = -360
\n" ); document.write( "(-4)*(90) = -360
\n" ); document.write( "(-5)*(72) = -360
\n" ); document.write( "(-6)*(60) = -360
\n" ); document.write( "(-8)*(45) = -360
\n" ); document.write( "(-9)*(40) = -360
\n" ); document.write( "(-10)*(36) = -360
\n" ); document.write( "(-12)*(30) = -360
\n" ); document.write( "(-15)*(24) = -360
\n" ); document.write( "(-18)*(20) = -360\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 $\"31\"$ :\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	-360	1+(-360)=-359
2	-180	2+(-180)=-178
3	-120	3+(-120)=-117
4	-90	4+(-90)=-86
5	-72	5+(-72)=-67
6	-60	6+(-60)=-54
8	-45	8+(-45)=-37
9	-40	9+(-40)=-31
10	-36	10+(-36)=-26
12	-30	12+(-30)=-18
15	-24	15+(-24)=-9
18	-20	18+(-20)=-2
-1	360	-1+360=359
-2	180	-2+180=178
-3	120	-3+120=117
-4	90	-4+90=86
-5	72	-5+72=67
-6	60	-6+60=54
-8	45	-8+45=37
-9	40	-9+40=31
-10	36	-10+36=26
-12	30	-12+30=18
-15	24	-15+24=9
-18	20	-18+20=2

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-9\"$ and $\"40\"$ add to $\"31\"$ (the middle coefficient).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-9\"$ and $\"40\"$ both multiply to $\"-360\"$ and add to $\"31\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"31v\"$ with $\"-9v%2B40v\"$ . Remember, $\"-9\"$ and $\"40\"$ add to $\"31\"$ . So this shows us that $\"-9v%2B40v=31v\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"5v%5E2%2Bhighlight%28-9v%2B40v%29-72\"$ Replace the second term $\"31v\"$ with $\"-9v%2B40v\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%285v%5E2-9v%29%2B%2840v-72%29\"$ Group the terms into two pairs.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"v%285v-9%29%2B%2840v-72%29\"$ Factor out the GCF $\"v\"$ from the first group.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"v%285v-9%29%2B8%285v-9%29\"$ Factor out $\"8\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"%28v%2B8%29%285v-9%29\"$ Combine like terms. Or factor out the common term $\"5v-9\"$ \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( "So $\"6%285v%5E2%2B31v-72%29\"$ then factors further to $\"6%28v%2B8%29%285v-9%29\"$ \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( "Answer:\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "So $\"30v%5E2%2B186v-432\"$ completely factors to $\"6%28v%2B8%29%285v-9%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"30v%5E2%2B186v-432=6%28v%2B8%29%285v-9%29\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"6%28v%2B8%29%285v-9%29\"$ to get $\"30v%5E2%2B186v-432\"$ or by graphing the original expression and the answer (the two graphs should be identical).
\n" ); document.write( " \n" ); document.write( "

\n" );