document.write( "Question 645131: how do i factor 6x^3 + 15x^2 + -9x \n" ); document.write( "

Algebra.Com's Answer #405372 by jim_thompson5910(35256) $\"\"$ $\"About$

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Note:\r
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\n" ); document.write( "\n" ); document.write( "6x^3 + 15x^2 + -9x\r
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\n" ); document.write( "\n" ); document.write( "6x^3 + 15x^2 - 9x\r
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\n" ); document.write( "\n" ); document.write( " $\"6x%5E3%2B15x%5E2-9x\"$ Start with the given expression.\r
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\n" ); document.write( "\n" ); document.write( " $\"3x%282x%5E2%2B5x-3%29\"$ Factor out the GCF $\"3x\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now let's try to factor the inner expression $\"2x%5E2%2B5x-3\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"2x%5E2%2B5x-3\"$ , 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
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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 $\"5x\"$ with $\"-x%2B6x\"$ . Remember, $\"-1\"$ and $\"6\"$ add to $\"5\"$ . So this shows us that $\"-x%2B6x=5x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"2x%5E2%2Bhighlight%28-x%2B6x%29-3\"$ Replace the second term $\"5x\"$ with $\"-x%2B6x\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%282x%5E2-x%29%2B%286x-3%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%282x-1%29%2B%286x-3%29\"$ Factor out the GCF $\"x\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"x%282x-1%29%2B3%282x-1%29\"$ 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( " $\"%28x%2B3%29%282x-1%29\"$ Combine like terms. Or factor out the common term $\"2x-1\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"3x%282x%5E2%2B5x-3%29\"$ then factors further to $\"3x%28x%2B3%29%282x-1%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"6x%5E3%2B15x%5E2-9x\"$ completely factors to $\"3x%28x%2B3%29%282x-1%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"6x%5E3%2B15x%5E2-9x=3x%28x%2B3%29%282x-1%29\"$ for all values of x.\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"3x%28x%2B3%29%282x-1%29\"$ to get $\"6x%5E3%2B15x%5E2-9x\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\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, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you\r
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