document.write( "Question 117565: r^3+7r^2-18r\r
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Algebra.Com's Answer #85551 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"r%5E3%2B7r%5E2-18r\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r%5E2%2B7r-18%29\"$ Factor out the GCF $\"r\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"r%5E2%2B7r-18\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"r%5E2%2B7r-18\"$ we can see that the first term is $\"r%5E2\"$ and the last term is $\"-18\"$ where the coefficients are 1 and -18 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -18 to get -18. Now what two numbers multiply to -18 and add to the middle coefficient 7? Let's list all of the factors of -18:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -18:\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-9,-18 ...List the negative factors as well. 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 -18\r
\n" ); document.write( "\n" ); document.write( "(1)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-9)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-6)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(18)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(9)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(6)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7\r
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First Number	Second Number	Sum
1	-18	1+(-18)=-17
2	-9	2+(-9)=-7
3	-6	3+(-6)=-3
-1	18	-1+18=17
-2	9	-2+9=7
-3	6	-3+6=3

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -2 and 9 add up to 7 and multiply to -18\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"r%5E2%2B7r-18\"$ , replace $\"7r\"$ with $\"-2r%2B9r\"$ (notice $\"-2r%2B9r\"$ adds up to $\"7r\"$ . So it is equivalent to $\"7r\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"r%5E2%2Bhighlight%28-2r%2B9r%29%2B-18\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"r%5E2-2r%2B9r-18\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%28r%5E2-2r%29%2B%289r-18%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r-2%29%2B9%28r-2%29\"$ Factor out the GCF of $\"r\"$ out of the first group. Factor out the GCF of $\"9\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28r%2B9%29%28r-2%29\"$ Since we have a common term of $\"r-2\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"r%5E2%2B7r-18\"$ factors to $\"%28r%2B9%29%28r-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r%2B9%29%28r-2%29\"$ Now reintroduce the GCF $\"r\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"r%5E3%2B7r%5E2-18r\"$ factors to $\"r%28r%2B9%29%28r-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"x%5E2y-9xy%5E2-36y%5E3\"$ we can see that the first term is $\"x%5E2y\"$ and the last term is $\"-36y%5E3\"$ where the coefficients are 1 and -36 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -36 to get -36. Now what two numbers multiply to -36 and add to the middle coefficient -9? Let's list all of the factors of -36:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -36:\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-9,-12,-18 ...List the negative factors as well. 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 -36\r
\n" ); document.write( "\n" ); document.write( "(1)*(-36)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-9)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(36)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(18)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(12)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(9)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to -9? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -9\r
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First Number	Second Number	Sum
1	-36	1+(-36)=-35
2	-18	2+(-18)=-16
3	-12	3+(-12)=-9
4	-9	4+(-9)=-5
-1	36	-1+36=35
-2	18	-2+18=16
-3	12	-3+12=9
-4	9	-4+9=5

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 3 and -12 add up to -9 and multiply to -36\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"x%5E2y-9xy%5E2-36y%5E3\"$ , replace $\"-9xy%5E2\"$ with $\"3xy%5E2%2B-12xy%5E2\"$ (notice $\"3xy%5E2%2B-12xy%5E2\"$ adds up to $\"-9xy%5E2\"$ . So it is equivalent to $\"-9xy%5E2\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"x%5E2y%2Bhighlight%283xy%5E2%2B-12xy%5E2%29%2B-36y%5E3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"x%5E2y%2B3xy%5E2-12xy%5E2-36y%5E3\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%28x%5E2y%2B3xy%5E2%29%2B%28-12xy%5E2-36y%5E3%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"xy%28x%2B3y%29-12y%5E2%28x%2B3y%29\"$ Factor out the GCF of $\"xy\"$ out of the first group. Factor out the GCF of $\"-12y%5E2\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28xy-12y%5E2%29%28x%2B3y%29\"$ Since we have a common term of $\"x%2B3y\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2y%2B3xy%5E2-12xy%5E2-36y%5E3\"$ factors to $\"%28xy-12y%5E2%29%28x%2B3y%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"x%5E2y-9xy%5E2-36y%5E3\"$ factors to $\"%28xy-12y%5E2%29%28x%2B3y%29\"$ (since $\"x%5E2y-9xy%5E2-36y%5E3\"$ is equivalent to $\"x%5E2y%2B3xy%5E2-12xy%5E2-36y%5E3\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So $\"x%5E2y-9xy%5E2-36y%5E3\"$ factors to $\"%28xy-12y%5E2%29%28x%2B3y%29\"$ \r
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