document.write( "Question 317059: Factor the following trinomial, if possible. If the coefficient of the first term is negative, factor out -1 to make the first term positive.
\n" ); document.write( "10a^2 - 9a + 2 \n" ); document.write( "

Algebra.Com's Answer #226994 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"10a%5E2-9a%2B2\"$ , we can see that the first coefficient is $\"10\"$ , the second coefficient is $\"-9\"$ , and the last term is $\"2\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"10\"$ by the last term $\"2\"$ to get $\"%2810%29%282%29=20\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"20\"$ (the previous product) and add to the second coefficient $\"-9\"$ ?\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 $\"20\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"20\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-10,-20\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 $\"20\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*20 = 20
\n" ); document.write( "2*10 = 20
\n" ); document.write( "4*5 = 20
\n" ); document.write( "(-1)*(-20) = 20
\n" ); document.write( "(-2)*(-10) = 20
\n" ); document.write( "(-4)*(-5) = 20\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 $\"-9\"$ :\r
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First Number	Second Number	Sum
1	20	1+20=21
2	10	2+10=12
4	5	4+5=9
-1	-20	-1+(-20)=-21
-2	-10	-2+(-10)=-12
-4	-5	-4+(-5)=-9

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-4\"$ and $\"-5\"$ add to $\"-9\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-4\"$ and $\"-5\"$ both multiply to $\"20\"$ and add to $\"-9\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-9a\"$ with $\"-4a-5a\"$ . Remember, $\"-4\"$ and $\"-5\"$ add to $\"-9\"$ . So this shows us that $\"-4a-5a=-9a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"10a%5E2%2Bhighlight%28-4a-5a%29%2B2\"$ Replace the second term $\"-9a\"$ with $\"-4a-5a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2810a%5E2-4a%29%2B%28-5a%2B2%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2a%285a-2%29%2B%28-5a%2B2%29\"$ Factor out the GCF $\"2a\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2a%285a-2%29-1%285a-2%29\"$ 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( " $\"%282a-1%29%285a-2%29\"$ Combine like terms. Or factor out the common term $\"5a-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"10a%5E2-9a%2B2\"$ factors to $\"%282a-1%29%285a-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"10a%5E2-9a%2B2=%282a-1%29%285a-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%282a-1%29%285a-2%29\"$ to get $\"10a%5E2-9a%2B2\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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