document.write( "Question 201026: factor the triominal
\n" ); document.write( "a^3 - 2a - 63a \n" ); document.write( "

Algebra.Com's Answer #151272 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
I'm going to assume you meant to say $\"a%5E3-2a%5E2-63a\"$ \r
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\n" ); document.write( "\n" ); document.write( " $\"a%5E3-2a%5E2-63a\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"a%28a%5E2-2a-63%29\"$ Factor out the GCF $\"a\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"a%5E2-2a-63\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"a%5E2-2a-63\"$ we can see that the first term is $\"a%5E2\"$ and the last term is $\"-63\"$ where the coefficients are 1 and -63 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 1 and the last coefficient -63 to get -63. Now what two numbers multiply to -63 and add to the middle coefficient -2? Let's list all of the factors of -63:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -63:\r
\n" ); document.write( "\n" ); document.write( "1,3,7,9,21,63\r
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\n" ); document.write( "\n" ); document.write( "-1,-3,-7,-9,-21,-63 ...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 -63\r
\n" ); document.write( "\n" ); document.write( "(1)*(-63)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-21)\r
\n" ); document.write( "\n" ); document.write( "(7)*(-9)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(63)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(21)\r
\n" ); document.write( "\n" ); document.write( "(-7)*(9)\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 -2? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to -2\r
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First Number	Second Number	Sum
1	-63	1+(-63)=-62
3	-21	3+(-21)=-18
7	-9	7+(-9)=-2
-1	63	-1+63=62
-3	21	-3+21=18
-7	9	-7+9=2

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\n" ); document.write( "\n" ); document.write( "From this list we can see that 7 and -9 add up to -2 and multiply to -63\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"a%5E2-2a-63\"$ , replace $\"-2a\"$ with $\"7a-9a\"$ (notice $\"7a-9a\"$ adds up to $\"-2a\"$ . So it is equivalent to $\"-2a\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"a%5E2%2Bhighlight%287a-9a%29-63\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"a%5E2%2B7a-9a-63\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%28a%5E2%2B7a%29%2B%28-9a-63%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"a%28a%2B7%29-9%28a%2B7%29\"$ Factor out the GCF of $\"a\"$ out of the first group. Factor out the GCF of $\"-9\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28a-9%29%28a%2B7%29\"$ Since we have a common term of $\"a%2B7\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"a%5E2%2B7a-9a-63\"$ factors to $\"%28a-9%29%28a%2B7%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"a%5E2-2a-63\"$ factors to $\"%28a-9%29%28a%2B7%29\"$ (since $\"a%5E2-2a-63\"$ is equivalent to $\"a%5E2%2B7a-9a-63\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"a%28a%5E2-2a-63%29\"$ and factors further to $\"a%28a-9%29%28a%2B7%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"a%5E3-2a%5E2-63a\"$ completely factors to $\"a%28a-9%29%28a%2B7%29\"$ \r
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