document.write( "Question 207857: factor completely, factor GCF first
\n" ); document.write( "8t^4+75t^3+27t^2 \n" ); document.write( "

Algebra.Com's Answer #157234 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"8t%5E4%2B75t%5E3%2B27t%5E2\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"t%5E2%288t%5E2%2B75t%2B27%29\"$ Factor out the GCF $\"t%5E2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"8t%5E2%2B75t%2B27\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"8t%5E2%2B75t%2B27\"$ , we can see that the first coefficient is $\"8\"$ , the second coefficient is $\"75\"$ , and the last term is $\"27\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"8\"$ by the last term $\"27\"$ to get $\"%288%29%2827%29=216\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"216\"$ (the previous product) and add to the second coefficient $\"75\"$ ?\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 $\"216\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"216\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,9,12,18,24,27,36,54,72,108,216\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-9,-12,-18,-24,-27,-36,-54,-72,-108,-216\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 $\"216\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*216 = 216
\n" ); document.write( "2*108 = 216
\n" ); document.write( "3*72 = 216
\n" ); document.write( "4*54 = 216
\n" ); document.write( "6*36 = 216
\n" ); document.write( "8*27 = 216
\n" ); document.write( "9*24 = 216
\n" ); document.write( "12*18 = 216
\n" ); document.write( "(-1)*(-216) = 216
\n" ); document.write( "(-2)*(-108) = 216
\n" ); document.write( "(-3)*(-72) = 216
\n" ); document.write( "(-4)*(-54) = 216
\n" ); document.write( "(-6)*(-36) = 216
\n" ); document.write( "(-8)*(-27) = 216
\n" ); document.write( "(-9)*(-24) = 216
\n" ); document.write( "(-12)*(-18) = 216\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 $\"75\"$ :\r
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First Number	Second Number	Sum
1	216	1+216=217
2	108	2+108=110
3	72	3+72=75
4	54	4+54=58
6	36	6+36=42
8	27	8+27=35
9	24	9+24=33
12	18	12+18=30
-1	-216	-1+(-216)=-217
-2	-108	-2+(-108)=-110
-3	-72	-3+(-72)=-75
-4	-54	-4+(-54)=-58
-6	-36	-6+(-36)=-42
-8	-27	-8+(-27)=-35
-9	-24	-9+(-24)=-33
-12	-18	-12+(-18)=-30

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"72\"$ add to $\"75\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"72\"$ both multiply to $\"216\"$ and add to $\"75\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"75t\"$ with $\"3t%2B72t\"$ . Remember, $\"3\"$ and $\"72\"$ add to $\"75\"$ . So this shows us that $\"3t%2B72t=75t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"8t%5E2%2Bhighlight%283t%2B72t%29%2B27\"$ Replace the second term $\"75t\"$ with $\"3t%2B72t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%288t%5E2%2B3t%29%2B%2872t%2B27%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"t%288t%2B3%29%2B%2872t%2B27%29\"$ Factor out the GCF $\"t\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"t%288t%2B3%29%2B9%288t%2B3%29\"$ Factor out $\"9\"$ 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( " $\"%28t%2B9%29%288t%2B3%29\"$ Combine like terms. Or factor out the common term $\"8t%2B3\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"t%5E2%288t%5E2%2B75t%2B27%29\"$ then factors further to $\"t%5E2%28t%2B9%29%288t%2B3%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"8t%5E4%2B75t%5E3%2B27t%5E2\"$ completely factors to $\"t%5E2%28t%2B9%29%288t%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"8t%5E4%2B75t%5E3%2B27t%5E2=t%5E2%28t%2B9%29%288t%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"t%5E2%28t%2B9%29%288t%2B3%29\"$ to get $\"8t%5E4%2B75t%5E3%2B27t%5E2\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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