document.write( "Question 547991: Help Me Factor 5p2-p-18 \n" ); document.write( "

Algebra.Com's Answer #356675 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"5p%5E2-p-18\"$ , we can see that the first coefficient is $\"5\"$ , the second coefficient is $\"-1\"$ , and the last term is $\"-18\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"5\"$ by the last term $\"-18\"$ to get $\"%285%29%28-18%29=-90\"$ .\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 $\"-1\"$ ?\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 $\"-90\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-90\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,5,6,9,10,15,18,30,45,90\r
\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) = -90
\n" ); document.write( "2*(-45) = -90
\n" ); document.write( "3*(-30) = -90
\n" ); document.write( "5*(-18) = -90
\n" ); document.write( "6*(-15) = -90
\n" ); document.write( "9*(-10) = -90
\n" ); document.write( "(-1)*(90) = -90
\n" ); document.write( "(-2)*(45) = -90
\n" ); document.write( "(-3)*(30) = -90
\n" ); document.write( "(-5)*(18) = -90
\n" ); document.write( "(-6)*(15) = -90
\n" ); document.write( "(-9)*(10) = -90\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	-90	1+(-90)=-89
2	-45	2+(-45)=-43
3	-30	3+(-30)=-27
5	-18	5+(-18)=-13
6	-15	6+(-15)=-9
9	-10	9+(-10)=-1
-1	90	-1+90=89
-2	45	-2+45=43
-3	30	-3+30=27
-5	18	-5+18=13
-6	15	-6+15=9
-9	10	-9+10=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"9\"$ and $\"-10\"$ add to $\"-1\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"9\"$ and $\"-10\"$ both multiply to $\"-90\"$ and add to $\"-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-1p\"$ with $\"9p-10p\"$ . Remember, $\"9\"$ and $\"-10\"$ add to $\"-1\"$ . So this shows us that $\"9p-10p=-1p\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"5p%5E2%2Bhighlight%289p-10p%29-18\"$ Replace the second term $\"-1p\"$ with $\"9p-10p\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%285p%5E2%2B9p%29%2B%28-10p-18%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"p%285p%2B9%29%2B%28-10p-18%29\"$ Factor out the GCF $\"p\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"p%285p%2B9%29-2%285p%2B9%29\"$ 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( " $\"%28p-2%29%285p%2B9%29\"$ Combine like terms. Or factor out the common term $\"5p%2B9\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"5p%5E2-p-18\"$ factors to $\"%28p-2%29%285p%2B9%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"5p%5E2-p-18=%28p-2%29%285p%2B9%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28p-2%29%285p%2B9%29\"$ to get $\"5p%5E2-p-18\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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