document.write( "Question 192309: please help factor out.n^2-11n-80
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Algebra.Com's Answer #144336 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"n%5E2-11n-80\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-11\"$ , and the last term is $\"-80\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-80\"$ to get $\"%281%29%28-80%29=-80\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-80\"$ (the previous product) and add to the second coefficient $\"-11\"$ ?\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 $\"-80\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-80\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,5,8,10,16,20,40,80\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-5,-8,-10,-16,-20,-40,-80\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 $\"-80\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-80)
\n" ); document.write( "2*(-40)
\n" ); document.write( "4*(-20)
\n" ); document.write( "5*(-16)
\n" ); document.write( "8*(-10)
\n" ); document.write( "(-1)*(80)
\n" ); document.write( "(-2)*(40)
\n" ); document.write( "(-4)*(20)
\n" ); document.write( "(-5)*(16)
\n" ); document.write( "(-8)*(10)\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 $\"-11\"$ :\r
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First Number	Second Number	Sum
1	-80	1+(-80)=-79
2	-40	2+(-40)=-38
4	-20	4+(-20)=-16
5	-16	5+(-16)=-11
8	-10	8+(-10)=-2
-1	80	-1+80=79
-2	40	-2+40=38
-4	20	-4+20=16
-5	16	-5+16=11
-8	10	-8+10=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"5\"$ and $\"-16\"$ add to $\"-11\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"5\"$ and $\"-16\"$ both multiply to $\"-80\"$ and add to $\"-11\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"-11n\"$ with $\"5n-16n\"$ . Remember, $\"5\"$ and $\"-16\"$ add to $\"-11\"$ . So this shows us that $\"5n-16n=-11n\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"n%5E2%2Bhighlight%285n-16n%29-80\"$ Replace the second term $\"-11n\"$ with $\"5n-16n\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28n%5E2%2B5n%29%2B%28-16n-80%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"n%28n%2B5%29%2B%28-16n-80%29\"$ Factor out the GCF $\"n\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"n%28n%2B5%29-16%28n%2B5%29\"$ Factor out $\"16\"$ 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( " $\"%28n-16%29%28n%2B5%29\"$ Combine like terms. Or factor out the common term $\"n%2B5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"n%5E2-11n-80\"$ factors to $\"%28n-16%29%28n%2B5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28n-16%29%28n%2B5%29\"$ to get $\"n%5E2-11n-80\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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