document.write( "Question 367308: How do you factor r^3+3r^2-54r \n" ); document.write( "

Algebra.Com's Answer #261816 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"r%5E3%2B3r%5E2-54r\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r%5E2%2B3r-54%29\"$ Factor out the GCF $\"r\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"r%5E2%2B3r-54\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"r%5E2%2B3r-54\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"3\"$ , and the last term is $\"-54\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-54\"$ to get $\"%281%29%28-54%29=-54\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-54\"$ (the previous product) and add to the second coefficient $\"3\"$ ?\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 $\"-54\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-54\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,9,18,27,54\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-9,-18,-27,-54\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 $\"-54\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-54) = -54
\n" ); document.write( "2*(-27) = -54
\n" ); document.write( "3*(-18) = -54
\n" ); document.write( "6*(-9) = -54
\n" ); document.write( "(-1)*(54) = -54
\n" ); document.write( "(-2)*(27) = -54
\n" ); document.write( "(-3)*(18) = -54
\n" ); document.write( "(-6)*(9) = -54\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 $\"3\"$ :\r
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First Number	Second Number	Sum
1	-54	1+(-54)=-53
2	-27	2+(-27)=-25
3	-18	3+(-18)=-15
6	-9	6+(-9)=-3
-1	54	-1+54=53
-2	27	-2+27=25
-3	18	-3+18=15
-6	9	-6+9=3

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-6\"$ and $\"9\"$ add to $\"3\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-6\"$ and $\"9\"$ both multiply to $\"-54\"$ and add to $\"3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"3r\"$ with $\"-6r%2B9r\"$ . Remember, $\"-6\"$ and $\"9\"$ add to $\"3\"$ . So this shows us that $\"-6r%2B9r=3r\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"r%5E2%2Bhighlight%28-6r%2B9r%29-54\"$ Replace the second term $\"3r\"$ with $\"-6r%2B9r\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28r%5E2-6r%29%2B%289r-54%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r-6%29%2B%289r-54%29\"$ Factor out the GCF $\"r\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r-6%29%2B9%28r-6%29\"$ Factor out $\"9\"$ 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( " $\"%28r%2B9%29%28r-6%29\"$ Combine like terms. Or factor out the common term $\"r-6\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"r%28r%5E2%2B3r-54%29\"$ then factors further to $\"r%28r%2B9%29%28r-6%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"r%5E3%2B3r%5E2-54r\"$ completely factors to $\"r%28r%2B9%29%28r-6%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"r%5E3%2B3r%5E2-54r=r%28r%2B9%29%28r-6%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"r%28r%2B9%29%28r-6%29\"$ to get $\"r%5E3%2B3r%5E2-54r\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
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\n" ); document.write( "\n" ); document.write( "Also, feel free to check out my tutoring website\r
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\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

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