document.write( "Question 550525: Factor completely, if prime, state this\r
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Algebra.Com's Answer #358827 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"5t%5E3-21t%5E2%2B18t\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"t%285t%5E2-21t%2B18%29\"$ Factor out the GCF $\"t\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"5t%5E2-21t%2B18\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"5t%5E2-21t%2B18\"$ , we can see that the first coefficient is $\"5\"$ , the second coefficient is $\"-21\"$ , and the last term is $\"18\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"5\"$ by the last term $\"18\"$ to get $\"%285%29%2818%29=90\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"90\"$ (the previous product) and add to the second coefficient $\"-21\"$ ?\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 $\"90\"$ (the previous product).\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
\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( "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 $\"90\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*90 = 90
\n" ); document.write( "2*45 = 90
\n" ); document.write( "3*30 = 90
\n" ); document.write( "5*18 = 90
\n" ); document.write( "6*15 = 90
\n" ); document.write( "9*10 = 90
\n" ); document.write( "(-1)*(-90) = 90
\n" ); document.write( "(-2)*(-45) = 90
\n" ); document.write( "(-3)*(-30) = 90
\n" ); document.write( "(-5)*(-18) = 90
\n" ); document.write( "(-6)*(-15) = 90
\n" ); document.write( "(-9)*(-10) = 90\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 $\"-21\"$ :\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 the table, we can see that the two numbers $\"-6\"$ and $\"-15\"$ add to $\"-21\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-6\"$ and $\"-15\"$ both multiply to $\"90\"$ and add to $\"-21\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-21t\"$ with $\"-6t-15t\"$ . Remember, $\"-6\"$ and $\"-15\"$ add to $\"-21\"$ . So this shows us that $\"-6t-15t=-21t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"5t%5E2%2Bhighlight%28-6t-15t%29%2B18\"$ Replace the second term $\"-21t\"$ with $\"-6t-15t\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%285t%5E2-6t%29%2B%28-15t%2B18%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"t%285t-6%29%2B%28-15t%2B18%29\"$ Factor out the GCF $\"t\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"t%285t-6%29-3%285t-6%29\"$ Factor out $\"3\"$ 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( " $\"%28t-3%29%285t-6%29\"$ Combine like terms. Or factor out the common term $\"5t-6\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"t%285t%5E2-21t%2B18%29\"$ then factors further to $\"t%28t-3%29%285t-6%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"5t%5E3-21t%5E2%2B18t\"$ completely factors to $\"t%28t-3%29%285t-6%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"5t%5E3-21t%5E2%2B18t=t%28t-3%29%285t-6%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"t%28t-3%29%285t-6%29\"$ to get $\"5t%5E3-21t%5E2%2B18t\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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