document.write( "Question 561415: Factor : x2+21x+54 \n" ); document.write( "

Algebra.Com's Answer #364270 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"x%5E2%2B21x%2B54\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"21\"$ , and the last term is $\"54\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"54\"$ to get $\"%281%29%2854%29=54\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"54\"$ (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 $\"54\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"54\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,9,18,27,54\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-9,-18,-27,-54\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 $\"54\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*54 = 54
\n" ); document.write( "2*27 = 54
\n" ); document.write( "3*18 = 54
\n" ); document.write( "6*9 = 54
\n" ); document.write( "(-1)*(-54) = 54
\n" ); document.write( "(-2)*(-27) = 54
\n" ); document.write( "(-3)*(-18) = 54
\n" ); document.write( "(-6)*(-9) = 54\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	54	1+54=55
2	27	2+27=29
3	18	3+18=21
6	9	6+9=15
-1	-54	-1+(-54)=-55
-2	-27	-2+(-27)=-29
-3	-18	-3+(-18)=-21
-6	-9	-6+(-9)=-15

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"3\"$ and $\"18\"$ add to $\"21\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"18\"$ both multiply to $\"54\"$ and add to $\"21\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"21x\"$ with $\"3x%2B18x\"$ . Remember, $\"3\"$ and $\"18\"$ add to $\"21\"$ . So this shows us that $\"3x%2B18x=21x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2%2Bhighlight%283x%2B18x%29%2B54\"$ Replace the second term $\"21x\"$ with $\"3x%2B18x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2%2B3x%29%2B%2818x%2B54%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B3%29%2B%2818x%2B54%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%28x%2B3%29%2B18%28x%2B3%29\"$ Factor out $\"18\"$ 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( " $\"%28x%2B18%29%28x%2B3%29\"$ Combine like terms. Or factor out the common term $\"x%2B3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2%2B21x%2B54\"$ factors to $\"%28x%2B18%29%28x%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"x%5E2%2B21x%2B54=%28x%2B18%29%28x%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B18%29%28x%2B3%29\"$ to get $\"x%5E2%2B21x%2B54\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
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\n" ); document.write( "\n" ); document.write( "Jim\r
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