document.write( "Question 221874: I need to understand how to do this problem by factoring completely. I believe my first step is finding the common factors. Can someone show me step by step?\r
\n" ); document.write( "\n" ); document.write( "27y^5 - 63y^4 + 30y^3\r
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Algebra.Com's Answer #166297 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
# 1\r
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\n" ); document.write( "\n" ); document.write( " $\"27y%5E5-63y%5E4%2B30y%5E3\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"3y%5E3%289y%5E2-21y%2B10%29\"$ Factor out the GCF $\"3y%5E3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"9y%5E2-21y%2B10\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"9y%5E2-21y%2B10\"$ we can see that the first term is $\"9y%5E2\"$ and the last term is $\"10\"$ where the coefficients are 9 and 10 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 9 and the last coefficient 10 to get 90. Now what two numbers multiply to 90 and add to the middle coefficient -21? Let's list all of the factors of 90:\r
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\n" ); document.write( "\n" ); document.write( "Factors of 90:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,5,6,9,10,15,18,30,45,90\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-9,-10,-15,-18,-30,-45,-90 ...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 90\r
\n" ); document.write( "\n" ); document.write( "1*90\r
\n" ); document.write( "\n" ); document.write( "2*45\r
\n" ); document.write( "\n" ); document.write( "3*30\r
\n" ); document.write( "\n" ); document.write( "5*18\r
\n" ); document.write( "\n" ); document.write( "6*15\r
\n" ); document.write( "\n" ); document.write( "9*10\r
\n" ); document.write( "\n" ); document.write( "(-1)*(-90)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(-45)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(-15)\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 -21? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -21\r
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First Number	Second Number	Sum
1	90	1+90=91
2	45	2+45=47
3	30	3+30=33
5	18	5+18=23
6	15	6+15=21
9	10	9+10=19
-1	-90	-1+(-90)=-91
-2	-45	-2+(-45)=-47
-3	-30	-3+(-30)=-33
-5	-18	-5+(-18)=-23
-6	-15	-6+(-15)=-21
-9	-10	-9+(-10)=-19

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -6 and -15 add up to -21 and multiply to 90\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"9y%5E2-21y%2B10\"$ , replace $\"-21y\"$ with $\"-6y%2B-15y\"$ (notice $\"-6y%2B-15y\"$ adds up to $\"-21y\"$ . So it is equivalent to $\"-21y\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"9y%5E2%2Bhighlight%28-6y%2B-15y%29%2B10\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"9y%5E2-6y-15y%2B10\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%289y%5E2-6y%29%2B%28-15y%2B10%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"3y%283y-2%29-5%283y-2%29\"$ Factor out the GCF of $\"3y\"$ out of the first group. Factor out the GCF of $\"-5\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%283y-5%29%283y-2%29\"$ Since we have a common term of $\"3y-2\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"9y%5E2-6y-15y%2B10\"$ factors to $\"%283y-5%29%283y-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"9y%5E2-21y%2B10\"$ factors to $\"%283y-5%29%283y-2%29\"$ (since $\"9y%5E2-21y%2B10\"$ is equivalent to $\"9y%5E2-6y-15y%2B10\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"3y%5E3%289y%5E2-21y%2B10%29\"$ and factors further to $\"3y%5E3%283y-5%29%283y-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"27y%5E5-63y%5E4%2B30y%5E3\"$ completely factors to $\"3y%5E3%283y-5%29%283y-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"27y%5E5-63y%5E4%2B30y%5E3=3y%5E3%283y-5%29%283y-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "# 2\r
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\n" ); document.write( "\n" ); document.write( " $\"125-8u%5E3\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"%285%29%5E3-%282u%29%5E3\"$ Rewrite $\"125\"$ as $\"%285%29%5E3\"$ . Rewrite $\"8u%5E3\"$ as $\"%282u%29%5E3\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%285-2u%29%28%285%29%5E2%2B%285%29%282u%29%2B%282u%29%5E2%29\"$ Now factor by using the difference of cubes formula. Remember the difference of cubes formula is $\"A%5E3-B%5E3=%28A-B%29%28A%5E2%2BAB%2BB%5E2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%285-2u%29%2825%2B10u%2B4u%5E2%29\"$ Multiply\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
\n" ); document.write( "\n" ); document.write( "So $\"125-8u%5E3\"$ factors to $\"%285-2u%29%2825%2B10u%2B4u%5E2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"125-8u%5E3=%285-2u%29%2825%2B10u%2B4u%5E2%29\"$
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