document.write( "Question 302560: 6x^5-11x^4+30x^3\r
\n" ); document.write( "\n" ); document.write( "Thank you very much for your help, I need to factor completly. \n" ); document.write( "

Algebra.Com's Answer #216942 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
\r
\n" ); document.write( "\n" ); document.write( " $\"6x%5E5-11x%5E4%2B30x%5E3\"$ Start with the given expression.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " $\"x%5E3%286x%5E2-11x%2B30%29\"$ Factor out the GCF $\"x%5E3\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"6x%5E2-11x%2B30\"$ \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 $\"6x%5E2-11x%2B30\"$ , we can see that the first coefficient is $\"6\"$ , the second coefficient is $\"-11\"$ , and the last term is $\"30\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"6\"$ by the last term $\"30\"$ to get $\"%286%29%2830%29=180\"$ .\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"180\"$ (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 $\"180\"$ (the previous product).\r
\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( "-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( "\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 $\"180\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*180 = 180
\n" ); document.write( "2*90 = 180
\n" ); document.write( "3*60 = 180
\n" ); document.write( "4*45 = 180
\n" ); document.write( "5*36 = 180
\n" ); document.write( "6*30 = 180
\n" ); document.write( "9*20 = 180
\n" ); document.write( "10*18 = 180
\n" ); document.write( "12*15 = 180
\n" ); document.write( "(-1)*(-180) = 180
\n" ); document.write( "(-2)*(-90) = 180
\n" ); document.write( "(-3)*(-60) = 180
\n" ); document.write( "(-4)*(-45) = 180
\n" ); document.write( "(-5)*(-36) = 180
\n" ); document.write( "(-6)*(-30) = 180
\n" ); document.write( "(-9)*(-20) = 180
\n" ); document.write( "(-10)*(-18) = 180
\n" ); document.write( "(-12)*(-15) = 180\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	180	1+180=181
2	90	2+90=92
3	60	3+60=63
4	45	4+45=49
5	36	5+36=41
6	30	6+30=36
9	20	9+20=29
10	18	10+18=28
12	15	12+15=27
-1	-180	-1+(-180)=-181
-2	-90	-2+(-90)=-92
-3	-60	-3+(-60)=-63
-4	-45	-4+(-45)=-49
-5	-36	-5+(-36)=-41
-6	-30	-6+(-30)=-36
-9	-20	-9+(-20)=-29
-10	-18	-10+(-18)=-28
-12	-15	-12+(-15)=-27

\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "From the table, we can see that there are no pairs of numbers which add to $\"-11\"$ . So $\"6x%5E2-11x%2B30\"$ cannot be factored.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "===============================================================\r
\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 $\"6x%5E5-11x%5E4%2B30x%5E3\"$ simply factors to $\"x%5E3%286x%5E2-11x%2B30%29\"$ \r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "In other words, $\"6x%5E5-11x%5E4%2B30x%5E3=x%5E3%286x%5E2-11x%2B30%29\"$ .
\n" ); document.write( " \n" ); document.write( "

\n" );