document.write( "Question 674303: Can someone walk me through the steps for the following problem? Thank you!\r
\n" ); document.write( "\n" ); document.write( "10a^2+25a-15\r
\n" ); document.write( "\n" ); document.write( "Factor completely. If a polynomial is prime, state this. \n" ); document.write( "

Algebra.Com's Answer #419146 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"10a%5E2%2B25a-15\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"5%282a%5E2%2B5a-3%29\"$ Factor out the GCF $\"5\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"2a%5E2%2B5a-3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"2a%5E2%2B5a-3\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"5\"$ , and the last term is $\"-3\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"2\"$ by the last term $\"-3\"$ to get $\"%282%29%28-3%29=-6\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-6\"$ (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 $\"-6\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-6\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6\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 $\"-6\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-6) = -6
\n" ); document.write( "2*(-3) = -6
\n" ); document.write( "(-1)*(6) = -6
\n" ); document.write( "(-2)*(3) = -6\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	-6	1+(-6)=-5
2	-3	2+(-3)=-1
-1	6	-1+6=5
-2	3	-2+3=1

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-1\"$ and $\"6\"$ add to $\"5\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-1\"$ and $\"6\"$ both multiply to $\"-6\"$ and add to $\"5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"5a\"$ with $\"-a%2B6a\"$ . Remember, $\"-1\"$ and $\"6\"$ add to $\"5\"$ . So this shows us that $\"-a%2B6a=5a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"2a%5E2%2Bhighlight%28-a%2B6a%29-3\"$ Replace the second term $\"5a\"$ with $\"-a%2B6a\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%282a%5E2-a%29%2B%286a-3%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"a%282a-1%29%2B%286a-3%29\"$ Factor out the GCF $\"a\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"a%282a-1%29%2B3%282a-1%29\"$ Factor out $\"3\"$ 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( " $\"%28a%2B3%29%282a-1%29\"$ Combine like terms. Or factor out the common term $\"2a-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"5%282a%5E2%2B5a-3%29\"$ then factors further to $\"5%28a%2B3%29%282a-1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"10a%5E2%2B25a-15\"$ completely factors to $\"5%28a%2B3%29%282a-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"10a%5E2%2B25a-15=5%28a%2B3%29%282a-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"5%28a%2B3%29%282a-1%29\"$ to get $\"10a%5E2%2B25a-15\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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