document.write( "Question 148757This question is from textbook
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Algebra.Com's Answer #109093 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"5x%5E2%2B16x%2B3\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"5x%5E2%2B16x%2B3\"$ , we can see that the first coefficient is $\"5\"$ , the second coefficient is $\"16\"$ , and the last term is $\"3\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"5\"$ by the last term $\"3\"$ to get $\"%285%29%283%29=15\"$ .\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 $\"16\"$ ?\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( "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 $\"15\"$ . For instance, $\"1%2A15=15\"$ , $\"3%2A5=15\"$ , etc.\r
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\n" ); document.write( "\n" ); document.write( "Since $\"15\"$ is positive, this means that either\r
\n" ); document.write( "\n" ); document.write( "a) both factors are positive, or...
\n" ); document.write( "b) both factors are negative.\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 $\"16\"$ :\r
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First Number	Second Number	Sum
1	15	1+15=16
3	5	3+5=8
-1	-15	-1+(-15)=-16
-3	-5	-3+(-5)=-8

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that the two numbers $\"1\"$ and $\"15\"$ add to $\"16\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"1\"$ and $\"15\"$ both multiply to $\"15\"$ and add to $\"16\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"16x\"$ with $\"x%2B15x\"$ . Remember, $\"1\"$ and $\"15\"$ add to $\"16\"$ . So this shows us that $\"x%2B15x=16x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"5x%5E2%2Bhighlight%28x%2B15x%29%2B3\"$ Replace the second term $\"16x\"$ with $\"x%2B15x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%285x%5E2%2Bx%29%2B%2815x%2B3%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%285x%2B1%29%2B%2815x%2B3%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%285x%2B1%29%2B3%285x%2B1%29\"$ 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( " $\"%28x%2B3%29%285x%2B1%29\"$ Combine like terms. Or factor out the common term $\"5x%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"5x%5E2%2B16x%2B3\"$ factors to $\"%28x%2B3%29%285x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by FOILing $\"%28x%2B3%29%285x%2B1%29\"$ to get $\"5x%5E2%2B16x%2B3\"$ or by graphing the original expression and the answer.
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