document.write( "Question 207386: factoring trinomials completely..........this is my problem and I can't figure out how to work any of them.....
\n" ); document.write( "36t^2+390t+64
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Algebra.Com's Answer #156814 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!

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\n" ); document.write( "\n" ); document.write( " $\"36t%5E2%2B390t%2B64\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"2%2818t%5E2%2B195t%2B32%29\"$ Factor out the GCF $\"2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"18t%5E2%2B195t%2B32\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"18t%5E2%2B195t%2B32\"$ , we can see that the first coefficient is $\"18\"$ , the second coefficient is $\"195\"$ , and the last term is $\"32\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"18\"$ by the last term $\"32\"$ to get $\"%2818%29%2832%29=576\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"576\"$ (the previous product) and add to the second coefficient $\"195\"$ ?\r
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\n" ); document.write( "\n" ); document.write( "To find these two numbers, we need to list all of the factors of $\"576\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"576\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,16,18,24,32,36,48,64,72,96,144,192,288,576\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-16,-18,-24,-32,-36,-48,-64,-72,-96,-144,-192,-288,-576\r
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\n" ); document.write( "\n" ); document.write( "Note: list the negative of each factor. 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 $\"576\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*576 = 576
\n" ); document.write( "2*288 = 576
\n" ); document.write( "3*192 = 576
\n" ); document.write( "4*144 = 576
\n" ); document.write( "6*96 = 576
\n" ); document.write( "8*72 = 576
\n" ); document.write( "9*64 = 576
\n" ); document.write( "12*48 = 576
\n" ); document.write( "16*36 = 576
\n" ); document.write( "18*32 = 576
\n" ); document.write( "24*24 = 576
\n" ); document.write( "(-1)*(-576) = 576
\n" ); document.write( "(-2)*(-288) = 576
\n" ); document.write( "(-3)*(-192) = 576
\n" ); document.write( "(-4)*(-144) = 576
\n" ); document.write( "(-6)*(-96) = 576
\n" ); document.write( "(-8)*(-72) = 576
\n" ); document.write( "(-9)*(-64) = 576
\n" ); document.write( "(-12)*(-48) = 576
\n" ); document.write( "(-16)*(-36) = 576
\n" ); document.write( "(-18)*(-32) = 576
\n" ); document.write( "(-24)*(-24) = 576\r
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\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 $\"195\"$ :\r
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First Number	Second Number	Sum
1	576	1+576=577
2	288	2+288=290
3	192	3+192=195
4	144	4+144=148
6	96	6+96=102
8	72	8+72=80
9	64	9+64=73
12	48	12+48=60
16	36	16+36=52
18	32	18+32=50
24	24	24+24=48
-1	-576	-1+(-576)=-577
-2	-288	-2+(-288)=-290
-3	-192	-3+(-192)=-195
-4	-144	-4+(-144)=-148
-6	-96	-6+(-96)=-102
-8	-72	-8+(-72)=-80
-9	-64	-9+(-64)=-73
-12	-48	-12+(-48)=-60
-16	-36	-16+(-36)=-52
-18	-32	-18+(-32)=-50
-24	-24	-24+(-24)=-48

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"192\"$ add to $\"195\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"192\"$ both multiply to $\"576\"$ and add to $\"195\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"195t\"$ with $\"3t%2B192t\"$ . Remember, $\"3\"$ and $\"192\"$ add to $\"195\"$ . So this shows us that $\"3t%2B192t=195t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"18t%5E2%2Bhighlight%283t%2B192t%29%2B32\"$ Replace the second term $\"195t\"$ with $\"3t%2B192t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2818t%5E2%2B3t%29%2B%28192t%2B32%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3t%286t%2B1%29%2B%28192t%2B32%29\"$ Factor out the GCF $\"3t\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3t%286t%2B1%29%2B32%286t%2B1%29\"$ Factor out $\"32\"$ from the second group. The goal of this step is to make the terms in the second parenthesis equal to the terms in the first parenthesis.\r
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\n" ); document.write( "\n" ); document.write( " $\"%283t%2B32%29%286t%2B1%29\"$ Combine like terms. Or factor out the common term $\"6t%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"2%2818t%5E2%2B195t%2B32%29\"$ then factors further to $\"2%283t%2B32%29%286t%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"36t%5E2%2B390t%2B64\"$ completely factors to $\"2%283t%2B32%29%286t%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"36t%5E2%2B390t%2B64=2%283t%2B32%29%286t%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"2%283t%2B32%29%286t%2B1%29\"$ to get $\"36t%5E2%2B390t%2B64\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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