document.write( "Question 369932: Factor completely: 6x^2+33x+15 \n" ); document.write( "

Algebra.Com's Answer #263579 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"6x%5E2%2B33x%2B15\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"3%282x%5E2%2B11x%2B5%29\"$ Factor out the GCF $\"3\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"2x%5E2%2B11x%2B5\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"2x%5E2%2B11x%2B5\"$ , we can see that the first coefficient is $\"2\"$ , the second coefficient is $\"11\"$ , and the last term is $\"5\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"2\"$ by the last term $\"5\"$ to get $\"%282%29%285%29=10\"$ .\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 $\"11\"$ ?\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
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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 $\"11\"$ :\r
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First Number	Second Number	Sum
1	10	1+10=11
2	5	2+5=7
-1	-10	-1+(-10)=-11
-2	-5	-2+(-5)=-7

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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 $\"11\"$ (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 $\"11\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"11x\"$ with $\"x%2B10x\"$ . Remember, $\"1\"$ and $\"10\"$ add to $\"11\"$ . So this shows us that $\"x%2B10x=11x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%5E2%2Bhighlight%28x%2B10x%29%2B5\"$ Replace the second term $\"11x\"$ with $\"x%2B10x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%282x%5E2%2Bx%29%2B%2810x%2B5%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%282x%2B1%29%2B%2810x%2B5%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%282x%2B1%29%2B5%282x%2B1%29\"$ 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( " $\"%28x%2B5%29%282x%2B1%29\"$ Combine like terms. Or factor out the common term $\"2x%2B1\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"3%282x%5E2%2B11x%2B5%29\"$ then factors further to $\"3%28x%2B5%29%282x%2B1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"6x%5E2%2B33x%2B15\"$ completely factors to $\"3%28x%2B5%29%282x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"6x%5E2%2B33x%2B15=3%28x%2B5%29%282x%2B1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"3%28x%2B5%29%282x%2B1%29\"$ to get $\"6x%5E2%2B33x%2B15\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\n" ); document.write( "If you need more help, email me at jim_thompson5910@hotmail.com\r
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\n" ); document.write( "\n" ); document.write( "Also, feel free to check out my tutoring website\r
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\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

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