document.write( "Question 112030: i was having trouble with this problem(Factor Completely)\r
\n" ); document.write( "\n" ); document.write( "5a^2-2ab-7b^2 \n" ); document.write( "

Algebra.Com's Answer #81716 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "Looking at $\"5a%5E2-2ab-7b%5E2\"$ we can see that the first term is $\"5a%5E2\"$ and the last term is $\"-7b%5E2\"$ where the coefficients are 5 and -7 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 5 and the last coefficient -7 to get -35. Now what two numbers multiply to -35 and add to the middle coefficient -2? Let's list all of the factors of -35:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -35:\r
\n" ); document.write( "\n" ); document.write( "1,5,7,35\r
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\n" ); document.write( "\n" ); document.write( "-1,-5,-7,-35 ...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 -35\r
\n" ); document.write( "\n" ); document.write( "(1)*(-35)\r
\n" ); document.write( "\n" ); document.write( "(5)*(-7)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(35)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(7)\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	-35	1+(-35)=-34
5	-7	5+(-7)=-2
-1	35	-1+35=34
-5	7	-5+7=2

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 5 and -7 add up to -2 and multiply to -35\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"5a%5E2-2ab-7b%5E2\"$ , replace $\"-2ab\"$ with $\"5ab%2B-7ab\"$ (notice $\"5ab%2B-7ab\"$ adds up to $\"-2ab\"$ . So it is equivalent to $\"-2ab\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"5a%5E2%2Bhighlight%285ab%2B-7ab%29%2B-7b%5E2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"5a%5E2%2B5ab-7ab-7b%5E2\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%285a%5E2%2B5ab%29%2B%28-7ab-7b%5E2%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"5a%28a%2Bb%29-7b%28a%2Bb%29\"$ Factor out the GCF of $\"5a\"$ out of the first group. Factor out the GCF of $\"-7b\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%285a-7b%29%28a%2Bb%29\"$ Since we have a common term of $\"a%2Bb\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"5a%5E2%2B5ab-7ab-7b%5E2\"$ factors to $\"%285a-7b%29%28a%2Bb%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"5a%5E2-2ab-7b%5E2\"$ factors to $\"%285a-7b%29%28a%2Bb%29\"$ (since $\"5a%5E2-2ab-7b%5E2\"$ is equivalent to $\"5a%5E2%2B5ab-7ab-7b%5E2\"$ )
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