document.write( "Question 197362: I need help on factoring a trinomial.\r
\n" ); document.write( "\n" ); document.write( " $\"r%5E3-2r%5E2-48r\"$ \n" ); document.write( "

Algebra.Com's Answer #147982 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!

\n" ); document.write( " $\"r%5E3-2r%5E2-48r\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r%5E2-2r-48%29\"$ Factor out the GCF $\"r\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"r%5E2-2r-48\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"r%5E2-2r-48\"$ we can see that the first term is $\"r%5E2\"$ and the last term is $\"-48\"$ where the coefficients are 1 and -48 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -48 to get -48. Now what two numbers multiply to -48 and add to the middle coefficient -2? Let's list all of the factors of -48:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -48:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,16,24,48\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-16,-24,-48 ...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 -48\r
\n" ); document.write( "\n" ); document.write( "(1)*(-48)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-16)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-8)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(48)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(24)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(16)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(12)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(8)\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 -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2\r
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First Number	Second Number	Sum
1	-48	1+(-48)=-47
2	-24	2+(-24)=-22
3	-16	3+(-16)=-13
4	-12	4+(-12)=-8
6	-8	6+(-8)=-2
-1	48	-1+48=47
-2	24	-2+24=22
-3	16	-3+16=13
-4	12	-4+12=8
-6	8	-6+8=2

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 6 and -8 add up to -2 and multiply to -48\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"r%5E2-2r-48\"$ , replace $\"-2r\"$ with $\"6r-8r\"$ (notice $\"6r-8r\"$ combines back to $\"-2r\"$ . So it is equivalent to $\"-2r\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"r%5E2%2Bhighlight%286r-8r%29-48\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"r%5E2%2B6r-8r-48\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%28r%5E2%2B6r%29%2B%28-8r-48%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"r%28r%2B6%29-8%28r%2B6%29\"$ Factor out the GCF of $\"r\"$ out of the first group. Factor out the GCF of $\"-8\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28r-8%29%28r%2B6%29\"$ Since we have a common term of $\"r%2B6\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"r%5E2%2B6r-8r-48\"$ factors to $\"%28r-8%29%28r%2B6%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"r%5E2-2r-48\"$ factors to $\"%28r-8%29%28r%2B6%29\"$ (since $\"r%5E2-2r-48\"$ is equivalent to $\"r%5E2%2B6r-8r-48\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"r%28r%5E2-2r-48%29\"$ and factors further to $\"r%28r-8%29%28r%2B6%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"r%5E3-2r%5E2-48r\"$ completely factors to $\"r%28r-8%29%28r%2B6%29\"$ \r
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