document.write( "Question 171356: Factoring Polynomials \r
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Algebra.Com's Answer #126498 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"20q%5E2%2B28q-24\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"4%285q%5E2%2B7q-6%29\"$ Factor out the GCF $\"4\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"5q%5E2%2B7q-6\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"5q%5E2%2B7q-6\"$ we can see that the first term is $\"5q%5E2\"$ and the last term is $\"-6\"$ where the coefficients are 5 and -6 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 5 and the last coefficient -6 to get -30. Now what two numbers multiply to -30 and add to the middle coefficient 7? Let's list all of the factors of -30:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -30:\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-5,-6,-10,-15,-30 ...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 -30\r
\n" ); document.write( "\n" ); document.write( "(1)*(-30)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-15)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-10)\r
\n" ); document.write( "\n" ); document.write( "(5)*(-6)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(30)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(15)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(10)\r
\n" ); document.write( "\n" ); document.write( "(-5)*(6)\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 7? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 7\r
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First Number	Second Number	Sum
1	-30	1+(-30)=-29
2	-15	2+(-15)=-13
3	-10	3+(-10)=-7
5	-6	5+(-6)=-1
-1	30	-1+30=29
-2	15	-2+15=13
-3	10	-3+10=7
-5	6	-5+6=1

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -3 and 10 add up to 7 and multiply to -30\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"5q%5E2%2B7q-6\"$ , replace $\"7q\"$ with $\"-3q%2B10q\"$ (notice $\"-3q%2B10q\"$ adds up to $\"7q\"$ . So it is equivalent to $\"7q\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"5q%5E2%2Bhighlight%28-3q%2B10q%29%2B-6\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"5q%5E2-3q%2B10q-6\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%285q%5E2-3q%29%2B%2810q-6%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"q%285q-3%29%2B2%285q-3%29\"$ Factor out the GCF of $\"q\"$ out of the first group. Factor out the GCF of $\"2\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28q%2B2%29%285q-3%29\"$ Since we have a common term of $\"5q-3\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"5q%5E2-3q%2B10q-6\"$ factors to $\"%28q%2B2%29%285q-3%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"5q%5E2%2B7q-6\"$ factors to $\"%28q%2B2%29%285q-3%29\"$ (since $\"5q%5E2%2B7q-6\"$ is equivalent to $\"5q%5E2-3q%2B10q-6\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"4%285q%5E2%2B7q-6%29\"$ and factors further to $\"4%28q%2B2%29%285q-3%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"20q%5E2%2B28q-24\"$ factors to $\"4%28q%2B2%29%285q-3%29\"$ \r
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