document.write( "Question 173210: Factorize 10x^2-x-3 \n" ); document.write( "

Algebra.Com's Answer #128058 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"10x%5E2-x-3\"$ , we can see that the first coefficient is $\"10\"$ , the second coefficient is $\"-1\"$ , and the last term is $\"-3\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"10\"$ by the last term $\"-3\"$ to get $\"%2810%29%28-3%29=-30\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-30\"$ (the previous product) and add to the second coefficient $\"-1\"$ ?\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 $\"-30\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-30\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,5,6,10,15,30\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-10,-15,-30\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 $\"-30\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-30)
\n" ); document.write( "2*(-15)
\n" ); document.write( "3*(-10)
\n" ); document.write( "5*(-6)
\n" ); document.write( "(-1)*(30)
\n" ); document.write( "(-2)*(15)
\n" ); document.write( "(-3)*(10)
\n" ); document.write( "(-5)*(6)\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 $\"-1\"$ :\r
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First Number	Second Number	Sum
1	-30	1+(-30)=-29
2	-15	2+(-15)=-13
3	-10	3+(-10)=-7
5	-6	5+(-6)=-1
-1	30	-1+30=29
-2	15	-2+15=13
-3	10	-3+10=7
-5	6	-5+6=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"5\"$ and $\"-6\"$ add to $\"-1\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"5\"$ and $\"-6\"$ both multiply to $\"-30\"$ and add to $\"-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-1x\"$ with $\"5x-6x\"$ . Remember, $\"5\"$ and $\"-6\"$ add to $\"-1\"$ . So this shows us that $\"5x-6x=-1x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"10x%5E2%2Bhighlight%285x-6x%29-3\"$ Replace the second term $\"-1x\"$ with $\"5x-6x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2810x%5E2%2B5x%29%2B%28-6x-3%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"5x%282x%2B1%29%2B%28-6x-3%29\"$ Factor out the GCF $\"5x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"5x%282x%2B1%29-3%282x%2B1%29\"$ Factor out $\"3\"$ 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( " $\"%285x-3%29%282x%2B1%29\"$ Combine like terms. Or factor out the common term $\"2x%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"10x%5E2-x-3\"$ factors to $\"%285x-3%29%282x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%285x-3%29%282x%2B1%29\"$ to get $\"10x%5E2-x-3\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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