document.write( "Question 347426: 21x^2-13x+2 \n" ); document.write( "

Algebra.Com's Answer #248391 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"21x%5E2-13x%2B2\"$ , we can see that the first coefficient is $\"21\"$ , the second coefficient is $\"-13\"$ , and the last term is $\"2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"21\"$ by the last term $\"2\"$ to get $\"%2821%29%282%29=42\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"42\"$ (the previous product) and add to the second coefficient $\"-13\"$ ?\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 $\"42\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"42\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,7,14,21,42\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-7,-14,-21,-42\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 $\"42\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*42 = 42
\n" ); document.write( "2*21 = 42
\n" ); document.write( "3*14 = 42
\n" ); document.write( "6*7 = 42
\n" ); document.write( "(-1)*(-42) = 42
\n" ); document.write( "(-2)*(-21) = 42
\n" ); document.write( "(-3)*(-14) = 42
\n" ); document.write( "(-6)*(-7) = 42\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 $\"-13\"$ :\r
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First Number	Second Number	Sum
1	42	1+42=43
2	21	2+21=23
3	14	3+14=17
6	7	6+7=13
-1	-42	-1+(-42)=-43
-2	-21	-2+(-21)=-23
-3	-14	-3+(-14)=-17
-6	-7	-6+(-7)=-13

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-6\"$ and $\"-7\"$ add to $\"-13\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-6\"$ and $\"-7\"$ both multiply to $\"42\"$ and add to $\"-13\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-13x\"$ with $\"-6x-7x\"$ . Remember, $\"-6\"$ and $\"-7\"$ add to $\"-13\"$ . So this shows us that $\"-6x-7x=-13x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"21x%5E2%2Bhighlight%28-6x-7x%29%2B2\"$ Replace the second term $\"-13x\"$ with $\"-6x-7x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2821x%5E2-6x%29%2B%28-7x%2B2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%287x-2%29%2B%28-7x%2B2%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%287x-2%29-1%287x-2%29\"$ Factor out $\"1\"$ 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( " $\"%283x-1%29%287x-2%29\"$ Combine like terms. Or factor out the common term $\"7x-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"21x%5E2-13x%2B2\"$ factors to $\"%283x-1%29%287x-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"21x%5E2-13x%2B2=%283x-1%29%287x-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%283x-1%29%287x-2%29\"$ to get $\"21x%5E2-13x%2B2\"$ 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( "Also, feel free to check out my tutoring website\r
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\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

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