document.write( "Question 120310: Factor the polymomial completely. If a ploynomial is prime say so.
\n" ); document.write( "9bn^3+15bn^2-14bn \n" ); document.write( "

Algebra.Com's Answer #88186 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( " $\"9bn%5E3%2B15bn%5E2-14bn\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"bn%289n%5E2%2B15n-14%29\"$ Factor out the GCF $\"bn\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"9n%5E2%2B15n-14\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"9n%5E2%2B15n-14\"$ we can see that the first term is $\"9n%5E2\"$ and the last term is $\"-14\"$ where the coefficients are 9 and -14 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 9 and the last coefficient -14 to get -126. Now what two numbers multiply to -126 and add to the middle coefficient 15? Let's list all of the factors of -126:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -126:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,6,7,9,14,18,21,42,63,126\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-6,-7,-9,-14,-18,-21,-42,-63,-126 ...List the negative factors as well. 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 -126\r
\n" ); document.write( "\n" ); document.write( "(1)*(-126)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-63)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-42)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-21)\r
\n" ); document.write( "\n" ); document.write( "(7)*(-18)\r
\n" ); document.write( "\n" ); document.write( "(9)*(-14)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(126)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(63)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(42)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(21)\r
\n" ); document.write( "\n" ); document.write( "(-7)*(18)\r
\n" ); document.write( "\n" ); document.write( "(-9)*(14)\r
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\n" ); document.write( "\n" ); document.write( "note: remember, the product of a negative and a positive number is a negative number\r
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\n" ); document.write( "\n" ); document.write( "Now which of these pairs add to 15? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 15\r
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First Number	Second Number	Sum
1	-126	1+(-126)=-125
2	-63	2+(-63)=-61
3	-42	3+(-42)=-39
6	-21	6+(-21)=-15
7	-18	7+(-18)=-11
9	-14	9+(-14)=-5
-1	126	-1+126=125
-2	63	-2+63=61
-3	42	-3+42=39
-6	21	-6+21=15
-7	18	-7+18=11
-9	14	-9+14=5

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -6 and 21 add up to 15 and multiply to -126\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"9n%5E2%2B15n-14\"$ , replace $\"15n\"$ with $\"-6n%2B21n\"$ (notice $\"-6n%2B21n\"$ adds up to $\"15n\"$ . So it is equivalent to $\"15n\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"9n%5E2%2Bhighlight%28-6n%2B21n%29%2B-14\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"9n%5E2-6n%2B21n-14\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%289n%5E2-6n%29%2B%2821n-14%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"3n%283n-2%29%2B7%283n-2%29\"$ Factor out the GCF of $\"3n\"$ out of the first group. Factor out the GCF of $\"7\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%283n%2B7%29%283n-2%29\"$ Since we have a common term of $\"3n-2\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"9n%5E2-6n%2B21n-14\"$ factors to $\"%283n%2B7%29%283n-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"9n%5E2%2B15n-14\"$ factors to $\"%283n%2B7%29%283n-2%29\"$ (since $\"9n%5E2%2B15n-14\"$ is equivalent to $\"9n%5E2-6n%2B21n-14\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"bn%289n%5E2%2B15n-14%29\"$ and factors further to $\"bn%283n%2B7%29%283n-2%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"9bn%5E3%2B15bn%5E2-14bn\"$ factors to $\"bn%283n%2B7%29%283n-2%29\"$ \r
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