document.write( "Question 355019: Factor w^2+5w-14 \n" ); document.write( "

Algebra.Com's Answer #253571 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"w%5E2%2B5w-14\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"5\"$ , and the last term is $\"-14\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-14\"$ to get $\"%281%29%28-14%29=-14\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-14\"$ (the previous product) and add to the second coefficient $\"5\"$ ?\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 $\"-14\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-14\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,7,14\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-7,-14\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 $\"-14\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-14) = -14
\n" ); document.write( "2*(-7) = -14
\n" ); document.write( "(-1)*(14) = -14
\n" ); document.write( "(-2)*(7) = -14\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 $\"5\"$ :\r
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First Number	Second Number	Sum
1	-14	1+(-14)=-13
2	-7	2+(-7)=-5
-1	14	-1+14=13
-2	7	-2+7=5

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-2\"$ and $\"7\"$ add to $\"5\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-2\"$ and $\"7\"$ both multiply to $\"-14\"$ and add to $\"5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"5w\"$ with $\"-2w%2B7w\"$ . Remember, $\"-2\"$ and $\"7\"$ add to $\"5\"$ . So this shows us that $\"-2w%2B7w=5w\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"w%5E2%2Bhighlight%28-2w%2B7w%29-14\"$ Replace the second term $\"5w\"$ with $\"-2w%2B7w\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28w%5E2-2w%29%2B%287w-14%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"w%28w-2%29%2B%287w-14%29\"$ Factor out the GCF $\"w\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"w%28w-2%29%2B7%28w-2%29\"$ 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( " $\"%28w%2B7%29%28w-2%29\"$ Combine like terms. Or factor out the common term $\"w-2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"w%5E2%2B5w-14\"$ factors to $\"%28w%2B7%29%28w-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"w%5E2%2B5w-14=%28w%2B7%29%28w-2%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"%28w%2B7%29%28w-2%29\"$ to get $\"w%5E2%2B5w-14\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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