document.write( "Question 319162: Factor completely. If the polynomial cannot be factored, enter PRIME.
\n" ); document.write( "Problem 7a2 + 63a - 70 \r
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Algebra.Com's Answer #228509 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"7a%5E2%2B63a-70\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"7%28a%5E2%2B9a-10%29\"$ Factor out the GCF $\"7\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"a%5E2%2B9a-10\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"a%5E2%2B9a-10\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"9\"$ , and the last term is $\"-10\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last term $\"-10\"$ to get $\"%281%29%28-10%29=-10\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-10\"$ (the previous product) and add to the second coefficient $\"9\"$ ?\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 $\"-10\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-10\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,5,10\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-5,-10\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 $\"-10\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-10) = -10
\n" ); document.write( "2*(-5) = -10
\n" ); document.write( "(-1)*(10) = -10
\n" ); document.write( "(-2)*(5) = -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 $\"9\"$ :\r
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First Number	Second Number	Sum
1	-10	1+(-10)=-9
2	-5	2+(-5)=-3
-1	10	-1+10=9
-2	5	-2+5=3

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-1\"$ and $\"10\"$ add to $\"9\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-1\"$ and $\"10\"$ both multiply to $\"-10\"$ and add to $\"9\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"9a\"$ with $\"-a%2B10a\"$ . Remember, $\"-1\"$ and $\"10\"$ add to $\"9\"$ . So this shows us that $\"-a%2B10a=9a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"a%5E2%2Bhighlight%28-a%2B10a%29-10\"$ Replace the second term $\"9a\"$ with $\"-a%2B10a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%28a%5E2-a%29%2B%2810a-10%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"a%28a-1%29%2B%2810a-10%29\"$ Factor out the GCF $\"a\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"a%28a-1%29%2B10%28a-1%29\"$ Factor out $\"10\"$ 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( " $\"%28a%2B10%29%28a-1%29\"$ Combine like terms. Or factor out the common term $\"a-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"7%28a%5E2%2B9a-10%29\"$ then factors further to $\"7%28a%2B10%29%28a-1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"7a%5E2%2B63a-70\"$ completely factors to $\"7%28a%2B10%29%28a-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"7a%5E2%2B63a-70=7%28a%2B10%29%28a-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"7%28a%2B10%29%28a-1%29\"$ to get $\"7a%5E2%2B63a-70\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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