document.write( "Question 132328: Factor completely:\r
\n" ); document.write( "\n" ); document.write( "5y^9 - 53y^6 + 72y^3. \r
\n" ); document.write( "\n" ); document.write( "I think that this problem has to do with cubing I am just not sure how to factor it. \n" ); document.write( "

Algebra.Com's Answer #96678 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"5y%5E9-53y%5E6%2B72y%5E3\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"y%5E3%285y%5E6-53y%5E3%2B72%29\"$ Factor out the GCF $\"y%5E3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"5y%5E6-53y%5E3%2B72\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"5y%5E6-53y%5E3%2B72\"$ we can see that the first term is $\"5y%5E6\"$ and the last term is $\"72\"$ where the coefficients are 5 and 72 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 5 and the last coefficient 72 to get 360. Now what two numbers multiply to 360 and add to the middle coefficient -53? Let's list all of the factors of 360:\r
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\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
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\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 ...List the negative factors as well. This will allow us to find all possible combinations\r
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\n" ); document.write( "\n" ); document.write( "These factors pair up and multiply to 360\r
\n" ); document.write( "\n" ); document.write( "1*360\r
\n" ); document.write( "\n" ); document.write( "2*180\r
\n" ); document.write( "\n" ); document.write( "3*120\r
\n" ); document.write( "\n" ); document.write( "4*90\r
\n" ); document.write( "\n" ); document.write( "5*72\r
\n" ); document.write( "\n" ); document.write( "6*60\r
\n" ); document.write( "\n" ); document.write( "8*45\r
\n" ); document.write( "\n" ); document.write( "9*40\r
\n" ); document.write( "\n" ); document.write( "10*36\r
\n" ); document.write( "\n" ); document.write( "12*30\r
\n" ); document.write( "\n" ); document.write( "15*24\r
\n" ); document.write( "\n" ); document.write( "18*20\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-360)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-180)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-120)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(-90)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-72)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-60)\r
\n" ); document.write( "\n" ); document.write( "(-8)*(-45)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(-40)\r
\n" ); document.write( "\n" ); document.write( "(-10)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(-12)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(-15)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(-18)*(-20)\r
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\n" ); document.write( "\n" ); document.write( "note: remember two negative numbers multiplied together make a positive number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -53? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -53\r
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First Number	Second Number	Sum
1	360	1+360=361
2	180	2+180=182
3	120	3+120=123
4	90	4+90=94
5	72	5+72=77
6	60	6+60=66
8	45	8+45=53
9	40	9+40=49
10	36	10+36=46
12	30	12+30=42
15	24	15+24=39
18	20	18+20=38
-1	-360	-1+(-360)=-361
-2	-180	-2+(-180)=-182
-3	-120	-3+(-120)=-123
-4	-90	-4+(-90)=-94
-5	-72	-5+(-72)=-77
-6	-60	-6+(-60)=-66
-8	-45	-8+(-45)=-53
-9	-40	-9+(-40)=-49
-10	-36	-10+(-36)=-46
-12	-30	-12+(-30)=-42
-15	-24	-15+(-24)=-39
-18	-20	-18+(-20)=-38

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -8 and -45 add up to -53 and multiply to 360\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"5y%5E6-53y%5E3%2B72\"$ , replace $\"-53y%5E3\"$ with $\"-8y%5E3%2B-45y%5E3\"$ (notice $\"-8y%5E3%2B-45y%5E3\"$ adds up to $\"-53y%5E3\"$ . So it is equivalent to $\"-53y%5E3\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"5y%5E6%2Bhighlight%28-8y%5E3%2B-45y%5E3%29%2B72\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"5y%5E6-8y%5E3-45y%5E3%2B72\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%285y%5E6-8y%5E3%29%2B%28-45y%5E3%2B72%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"y%5E3%285y%5E3-8%29-9%285y%5E3-8%29\"$ Factor out the GCF of $\"y%5E3\"$ out of the first group. Factor out the GCF of $\"-9\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28y%5E3-9%29%285y%5E3-8%29\"$ Since we have a common term of $\"5y%5E3-8\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"5y%5E6-8y%5E3-45y%5E3%2B72\"$ factors to $\"%28y%5E3-9%29%285y%5E3-8%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"5y%5E6-53y%5E3%2B72\"$ factors to $\"%28y%5E3-9%29%285y%5E3-8%29\"$ (since $\"5y%5E6-53y%5E3%2B72\"$ is equivalent to $\"5y%5E6-8y%5E3-45y%5E3%2B72\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"y%5E3%285y%5E6-53y%5E3%2B72%29\"$ and factors further to $\"y%5E3%28y%5E3-9%29%285y%5E3-8%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"5y%5E9-53y%5E6%2B72y%5E3\"$ factors to $\"y%5E3%28y%5E3-9%29%285y%5E3-8%29\"$ \r
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