document.write( "Question 551194: factor 22nsquare+n-5\r
\n" ); document.write( "\n" ); document.write( "also factor 14zsquare-49z+35\r
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\n" ); document.write( "\n" ); document.write( "And factor 5xsquare+17x+6 \n" ); document.write( "

Algebra.Com's Answer #359520 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'll do the first one to get you started\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"22n%5E2%2Bn-5\"$ , we can see that the first coefficient is $\"22\"$ , the second coefficient is $\"1\"$ , and the last term is $\"-5\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"22\"$ by the last term $\"-5\"$ to get $\"%2822%29%28-5%29=-110\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-110\"$ (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 $\"-110\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-110\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,5,10,11,22,55,110\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-5,-10,-11,-22,-55,-110\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 $\"-110\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-110) = -110
\n" ); document.write( "2*(-55) = -110
\n" ); document.write( "5*(-22) = -110
\n" ); document.write( "10*(-11) = -110
\n" ); document.write( "(-1)*(110) = -110
\n" ); document.write( "(-2)*(55) = -110
\n" ); document.write( "(-5)*(22) = -110
\n" ); document.write( "(-10)*(11) = -110\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	-110	1+(-110)=-109
2	-55	2+(-55)=-53
5	-22	5+(-22)=-17
10	-11	10+(-11)=-1
-1	110	-1+110=109
-2	55	-2+55=53
-5	22	-5+22=17
-10	11	-10+11=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-10\"$ and $\"11\"$ add to $\"1\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-10\"$ and $\"11\"$ both multiply to $\"-110\"$ and add to $\"1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"1n\"$ with $\"-10n%2B11n\"$ . Remember, $\"-10\"$ and $\"11\"$ add to $\"1\"$ . So this shows us that $\"-10n%2B11n=1n\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"22n%5E2%2Bhighlight%28-10n%2B11n%29-5\"$ Replace the second term $\"1n\"$ with $\"-10n%2B11n\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%2822n%5E2-10n%29%2B%2811n-5%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"2n%2811n-5%29%2B%2811n-5%29\"$ Factor out the GCF $\"2n\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"2n%2811n-5%29%2B1%2811n-5%29\"$ Factor out $\"1\"$ 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( " $\"%282n%2B1%29%2811n-5%29\"$ Combine like terms. Or factor out the common term $\"11n-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"22n%5E2%2Bn-5\"$ factors to $\"%282n%2B1%29%2811n-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"22n%5E2%2Bn-5=%282n%2B1%29%2811n-5%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%282n%2B1%29%2811n-5%29\"$ to get $\"22n%5E2%2Bn-5\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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