document.write( "Question 640411: Could you help me with this problem about factoring polynomials completely\r
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Algebra.Com's Answer #403223 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( " $\"12x%5E3%2B8x%5E2-20x\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"4x%283x%5E2%2B2x-5%29\"$ Factor out the GCF $\"4x\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"3x%5E2%2B2x-5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"3x%5E2%2B2x-5\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"2\"$ , and the last term is $\"-5\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"3\"$ by the last term $\"-5\"$ to get $\"%283%29%28-5%29=-15\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-15\"$ (the previous product) and add to the second coefficient $\"2\"$ ?\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 $\"-15\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"-15\"$ :\r
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\n" ); document.write( "\n" ); document.write( "-1,-3,-5,-15\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 $\"-15\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-15) = -15
\n" ); document.write( "3*(-5) = -15
\n" ); document.write( "(-1)*(15) = -15
\n" ); document.write( "(-3)*(5) = -15\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 $\"2\"$ :\r
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First Number	Second Number	Sum
1	-15	1+(-15)=-14
3	-5	3+(-5)=-2
-1	15	-1+15=14
-3	5	-3+5=2

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"-3\"$ and $\"5\"$ add to $\"2\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-3\"$ and $\"5\"$ both multiply to $\"-15\"$ and add to $\"2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"2x\"$ with $\"-3x%2B5x\"$ . Remember, $\"-3\"$ and $\"5\"$ add to $\"2\"$ . So this shows us that $\"-3x%2B5x=2x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%5E2%2Bhighlight%28-3x%2B5x%29-5\"$ Replace the second term $\"2x\"$ with $\"-3x%2B5x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%283x%5E2-3x%29%2B%285x-5%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%28x-1%29%2B%285x-5%29\"$ Factor out the GCF $\"3x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%28x-1%29%2B5%28x-1%29\"$ Factor out $\"5\"$ 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( " $\"%283x%2B5%29%28x-1%29\"$ Combine like terms. Or factor out the common term $\"x-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"4x%283x%5E2%2B2x-5%29\"$ then factors further to $\"4x%283x%2B5%29%28x-1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"12x%5E3%2B8x%5E2-20x\"$ completely factors to $\"4x%283x%2B5%29%28x-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"12x%5E3%2B8x%5E2-20x=4x%283x%2B5%29%28x-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"4x%283x%2B5%29%28x-1%29\"$ to get $\"12x%5E3%2B8x%5E2-20x\"$ or by graphing the original expression and the answer (the two graphs should be identical).
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