document.write( "Question 594532: Factor each polynomial.
\n" ); document.write( "n^2+n-42 \n" ); document.write( "

Algebra.Com's Answer #376786 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"n%5E2%2Bn-42\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"1\"$ , and the last term is $\"-42\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-42\"$ to get $\"%281%29%28-42%29=-42\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-42\"$ (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 $\"-42\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-42\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-7,-14,-21,-42\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 $\"-42\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-42) = -42
\n" ); document.write( "2*(-21) = -42
\n" ); document.write( "3*(-14) = -42
\n" ); document.write( "6*(-7) = -42
\n" ); document.write( "(-1)*(42) = -42
\n" ); document.write( "(-2)*(21) = -42
\n" ); document.write( "(-3)*(14) = -42
\n" ); document.write( "(-6)*(7) = -42\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	-42	1+(-42)=-41
2	-21	2+(-21)=-19
3	-14	3+(-14)=-11
6	-7	6+(-7)=-1
-1	42	-1+42=41
-2	21	-2+21=19
-3	14	-3+14=11
-6	7	-6+7=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-6\"$ and $\"7\"$ add to $\"1\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-6\"$ and $\"7\"$ both multiply to $\"-42\"$ and add to $\"1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"1n\"$ with $\"-6n%2B7n\"$ . Remember, $\"-6\"$ and $\"7\"$ add to $\"1\"$ . So this shows us that $\"-6n%2B7n=1n\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"n%5E2%2Bhighlight%28-6n%2B7n%29-42\"$ Replace the second term $\"1n\"$ with $\"-6n%2B7n\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28n%5E2-6n%29%2B%287n-42%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"n%28n-6%29%2B%287n-42%29\"$ Factor out the GCF $\"n\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"n%28n-6%29%2B7%28n-6%29\"$ Factor out $\"7\"$ 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( " $\"%28n%2B7%29%28n-6%29\"$ Combine like terms. Or factor out the common term $\"n-6\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"n%5E2%2Bn-42\"$ factors to $\"%28n%2B7%29%28n-6%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"n%5E2%2Bn-42=%28n%2B7%29%28n-6%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28n%2B7%29%28n-6%29\"$ to get $\"n%5E2%2Bn-42\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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