document.write( "Question 174855: What are the two binomial factors of 6s^2+40s-64? \n" ); document.write( "

Algebra.Com's Answer #129927 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( " $\"6s%5E2%2B40s-64\"$ Start with the given expression\r
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\n" ); document.write( "\n" ); document.write( " $\"2%283s%5E2%2B20s-32%29\"$ Factor out the GCF $\"2\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's focus on the inner expression $\"3s%5E2%2B20s-32\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at $\"3s%5E2%2B20s-32\"$ we can see that the first term is $\"3s%5E2\"$ and the last term is $\"-32\"$ where the coefficients are 3 and -32 respectively.\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient 3 and the last coefficient -32 to get -96. Now what two numbers multiply to -96 and add to the middle coefficient 20? Let's list all of the factors of -96:\r
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\n" ); document.write( "\n" ); document.write( "Factors of -96:\r
\n" ); document.write( "\n" ); document.write( "1,2,3,4,6,8,12,16,24,32,48,96\r
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\n" ); document.write( "\n" ); document.write( "-1,-2,-3,-4,-6,-8,-12,-16,-24,-32,-48,-96 ...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 -96\r
\n" ); document.write( "\n" ); document.write( "(1)*(-96)\r
\n" ); document.write( "\n" ); document.write( "(2)*(-48)\r
\n" ); document.write( "\n" ); document.write( "(3)*(-32)\r
\n" ); document.write( "\n" ); document.write( "(4)*(-24)\r
\n" ); document.write( "\n" ); document.write( "(6)*(-16)\r
\n" ); document.write( "\n" ); document.write( "(8)*(-12)\r
\n" ); document.write( "\n" ); document.write( "(-1)*(96)\r
\n" ); document.write( "\n" ); document.write( "(-2)*(48)\r
\n" ); document.write( "\n" ); document.write( "(-3)*(32)\r
\n" ); document.write( "\n" ); document.write( "(-4)*(24)\r
\n" ); document.write( "\n" ); document.write( "(-6)*(16)\r
\n" ); document.write( "\n" ); document.write( "(-8)*(12)\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 20? Lets make a table of all of the pairs of factors we multiplied and see which two numbers add to 20\r
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First Number	Second Number	Sum
1	-96	1+(-96)=-95
2	-48	2+(-48)=-46
3	-32	3+(-32)=-29
4	-24	4+(-24)=-20
6	-16	6+(-16)=-10
8	-12	8+(-12)=-4
-1	96	-1+96=95
-2	48	-2+48=46
-3	32	-3+32=29
-4	24	-4+24=20
-6	16	-6+16=10
-8	12	-8+12=4

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\n" ); document.write( "\n" ); document.write( "From this list we can see that -4 and 24 add up to 20 and multiply to -96\r
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\n" ); document.write( "\n" ); document.write( "Now looking at the expression $\"3s%5E2%2B20s-32\"$ , replace $\"20s\"$ with $\"-4s%2B24s\"$ (notice $\"-4s%2B24s\"$ adds up to $\"20s\"$ . So it is equivalent to $\"20s\"$ )\r
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\n" ); document.write( "\n" ); document.write( " $\"3s%5E2%2Bhighlight%28-4s%2B24s%29%2B-32\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now let's factor $\"3s%5E2-4s%2B24s-32\"$ by grouping:\r
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\n" ); document.write( "\n" ); document.write( " $\"%283s%5E2-4s%29%2B%2824s-32%29\"$ Group like terms\r
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\n" ); document.write( "\n" ); document.write( " $\"s%283s-4%29%2B8%283s-4%29\"$ Factor out the GCF of $\"s\"$ out of the first group. Factor out the GCF of $\"8\"$ out of the second group\r
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\n" ); document.write( "\n" ); document.write( " $\"%28s%2B8%29%283s-4%29\"$ Since we have a common term of $\"3s-4\"$ , we can combine like terms\r
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\n" ); document.write( "\n" ); document.write( "So $\"3s%5E2-4s%2B24s-32\"$ factors to $\"%28s%2B8%29%283s-4%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So this also means that $\"3s%5E2%2B20s-32\"$ factors to $\"%28s%2B8%29%283s-4%29\"$ (since $\"3s%5E2%2B20s-32\"$ is equivalent to $\"3s%5E2-4s%2B24s-32\"$ )\r
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\n" ); document.write( "\n" ); document.write( "So our expression goes from $\"2%283s%5E2%2B20s-32%29\"$ and factors further to $\"2%28s%2B8%29%283s-4%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"6s%5E2%2B40s-64\"$ factors to $\"2%28s%2B8%29%283s-4%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "So the two binomial factors are $\"s%2B8\"$ and $\"3s-4\"$ \n" ); document.write( "

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