document.write( "Question 152320: Factor:\r
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\n" ); document.write( "\n" ); document.write( "9x^2 - 21x + 10 \n" ); document.write( "

Algebra.Com's Answer #111987 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"9x%5E2-21x%2B10\"$ , we can see that the first coefficient is $\"9\"$ , the second coefficient is $\"-21\"$ , and the last term is $\"10\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"9\"$ by the last term $\"10\"$ to get $\"%289%29%2810%29=90\"$ .\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
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\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
\n" ); document.write( "2*45
\n" ); document.write( "3*30
\n" ); document.write( "5*18
\n" ); document.write( "6*15
\n" ); document.write( "9*10
\n" ); document.write( "(-1)*(-90)
\n" ); document.write( "(-2)*(-45)
\n" ); document.write( "(-3)*(-30)
\n" ); document.write( "(-5)*(-18)
\n" ); document.write( "(-6)*(-15)
\n" ); document.write( "(-9)*(-10)\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 $\"-21x\"$ with $\"-6x-15x\"$ . Remember, $\"-6\"$ and $\"-15\"$ add to $\"-21\"$ . So this shows us that $\"-6x-15x=-21x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"9x%5E2%2Bhighlight%28-6x-15x%29%2B10\"$ Replace the second term $\"-21x\"$ with $\"-6x-15x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%289x%5E2-6x%29%2B%28-15x%2B10%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%283x-2%29%2B%28-15x%2B10%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%283x-2%29-5%283x-2%29\"$ Factor out $\"5\"$ 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( " $\"%283x-5%29%283x-2%29\"$ Combine like terms. Or factor out the common term $\"3x-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"9x%5E2-21x%2B10\"$ factors to $\"%283x-5%29%283x-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%283x-5%29%283x-2%29\"$ to get $\"9x%5E2-21x%2B10\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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