document.write( "Question 131216: factor completely: 6(y-4)^2-(y-4)-35 \n" ); document.write( "

Algebra.Com's Answer #95818 by jim_thompson5910(35256) $\"\"$ $\"About$

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$\"6%28y-4%29%5E2-%28y-4%29-35\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( "Let $\"z=y-4\"$ . So our expression becomes\r
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\n" ); document.write( "\n" ); document.write( " $\"6z%5E2-z-35\"$ Plug in $\"z=y-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"6z%5E2-z-35\"$ we can see that the first term is $\"6z%5E2\"$ 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 -1? 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 -1? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -1\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

\r
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\n" ); document.write( "\n" ); document.write( "From this list we can see that 14 and -15 add up to -1 and multiply to -210\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"6z%5E2-z-35\"$ , replace $\"-z\"$ with $\"14z%2B-15z\"$ (notice $\"14z%2B-15z\"$ adds up to $\"-z\"$ . So it is equivalent to $\"-z\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"6z%5E2%2Bhighlight%2814z%2B-15z%29%2B-35\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"6z%5E2%2B14z-15z-35\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%286z%5E2%2B14z%29%2B%28-15z-35%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"2z%283z%2B7%29-5%283z%2B7%29\"$ Factor out the GCF of $\"2z\"$ 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( " $\"%282z-5%29%283z%2B7%29\"$ Since we have a common term of $\"3z%2B7\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"6z%5E2-z-35\"$ factors to $\"%282z-5%29%283z%2B7%29\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%282%28y-4%29-5%29%283%28y-4%29%2B7%29\"$ Now replace z with $\"y-4\"$ . Remember, we let $\"z=y-4\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"%282y-8-5%29%283y-12%2B7%29\"$ Distribute\r
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\n" ); document.write( "\n" ); document.write( " $\"%282y-13%29%283y-5%29\"$ Combine like terms\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"6%28y-4%29%5E2-%28y-4%29-35\"$ factors to $\"%282y-13%29%283y-5%29\"$ \n" ); document.write( "

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