document.write( "Question 131051: factor completely: 6y^3+11y^2-35y \n" ); document.write( "

Algebra.Com's Answer #95651 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"6y%5E3%2B11y%5E2-35y\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"y%286y%5E2%2B11y-35%29\"$ Factor out the GCF $\"y\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"6y%5E2%2B11y-35\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"6y%5E2%2B11y-35\"$ we can see that the first term is $\"6y%5E2\"$ and the last term is $\"-35\"$ where the coefficients are 6 and -35 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 6 and the last coefficient -35 to get -210. Now what two numbers multiply to -210 and add to the middle coefficient 11? Let's list all of the factors of -210:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -210:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,5,6,7,10,14,15,21,30,35,42,70,105,210\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-7,-10,-14,-15,-21,-30,-35,-42,-70,-105,-210 ...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 -210\r
\n" ); document.write( "\n" ); document.write( "(1)*(-210)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-105)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-70)\r
\n" ); document.write( "\n" ); document.write( "(5)*(-42)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-35)\r
\n" ); document.write( "\n" ); document.write( "(7)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(10)*(-21)\r
\n" ); document.write( "\n" ); document.write( "(14)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(210)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(105)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(70)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(42)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(35)\r
\n" ); document.write( "\n" ); document.write( "(-7)*(30)\r
\n" ); document.write( "\n" ); document.write( "(-10)*(21)\r
\n" ); document.write( "\n" ); document.write( "(-14)*(15)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 11? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 11\r
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First Number	Second Number	Sum
1	-210	1+(-210)=-209
2	-105	2+(-105)=-103
3	-70	3+(-70)=-67
5	-42	5+(-42)=-37
6	-35	6+(-35)=-29
7	-30	7+(-30)=-23
10	-21	10+(-21)=-11
14	-15	14+(-15)=-1
-1	210	-1+210=209
-2	105	-2+105=103
-3	70	-3+70=67
-5	42	-5+42=37
-6	35	-6+35=29
-7	30	-7+30=23
-10	21	-10+21=11
-14	15	-14+15=1

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -10 and 21 add up to 11 and multiply to -210\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"6y%5E2%2B11y-35\"$ , replace $\"11y\"$ with $\"-10y%2B21y\"$ (notice $\"-10y%2B21y\"$ adds up to $\"11y\"$ . So it is equivalent to $\"11y\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"6y%5E2%2Bhighlight%28-10y%2B21y%29%2B-35\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"6y%5E2-10y%2B21y-35\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%286y%5E2-10y%29%2B%2821y-35%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"2y%283y-5%29%2B7%283y-5%29\"$ Factor out the GCF of $\"2y\"$ out of the first group. Factor out the GCF of $\"7\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%282y%2B7%29%283y-5%29\"$ Since we have a common term of $\"3y-5\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"6y%5E2-10y%2B21y-35\"$ factors to $\"%282y%2B7%29%283y-5%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"6y%5E2%2B11y-35\"$ factors to $\"%282y%2B7%29%283y-5%29\"$ (since $\"6y%5E2%2B11y-35\"$ is equivalent to $\"6y%5E2-10y%2B21y-35\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"y%286y%5E2%2B11y-35%29\"$ and factors further to $\"y%282y%2B7%29%283y-5%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"6y%5E3%2B11y%5E2-35y\"$ factors to $\"y%282y%2B7%29%283y-5%29\"$ \r
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