document.write( "Question 561239: Please help me factor trinomials with the form axsquared + bx +c.
\n" ); document.write( " An example 7xsquared +17x+6. Any help is greatly appreciated. Joey \n" ); document.write( "

Algebra.Com's Answer #364214 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"7x%5E2%2B17x%2B6\"$ , we can see that the first coefficient is $\"7\"$ , the second coefficient is $\"17\"$ , and the last term is $\"6\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"7\"$ by the last term $\"6\"$ to get $\"%287%29%286%29=42\"$ .\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 $\"17\"$ ?\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 $\"17\"$ :\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 $\"3\"$ and $\"14\"$ add to $\"17\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"3\"$ and $\"14\"$ both multiply to $\"42\"$ and add to $\"17\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"17x\"$ with $\"3x%2B14x\"$ . Remember, $\"3\"$ and $\"14\"$ add to $\"17\"$ . So this shows us that $\"3x%2B14x=17x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"7x%5E2%2Bhighlight%283x%2B14x%29%2B6\"$ Replace the second term $\"17x\"$ with $\"3x%2B14x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%287x%5E2%2B3x%29%2B%2814x%2B6%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%287x%2B3%29%2B%2814x%2B6%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%287x%2B3%29%2B2%287x%2B3%29\"$ Factor out $\"2\"$ 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( " $\"%28x%2B2%29%287x%2B3%29\"$ Combine like terms. Or factor out the common term $\"7x%2B3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"7x%5E2%2B17x%2B6\"$ factors to $\"%28x%2B2%29%287x%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"7x%5E2%2B17x%2B6=%28x%2B2%29%287x%2B3%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28x%2B2%29%287x%2B3%29\"$ to get $\"7x%5E2%2B17x%2B6\"$ 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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