document.write( "Question 522895: Which of the following is one of the two binomial factors of 6s2 + 40s – 64? \n" ); document.write( "

Algebra.Com's Answer #347061 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 try to factor the inner expression $\"3s%5E2%2B20s-32\"$ \r
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"3s%5E2%2B20s-32\"$ , we can see that the first coefficient is $\"3\"$ , the second coefficient is $\"20\"$ , and the last term is $\"-32\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"3\"$ by the last term $\"-32\"$ to get $\"%283%29%28-32%29=-96\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"-96\"$ (the previous product) and add to the second coefficient $\"20\"$ ?\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 $\"-96\"$ (the previous product).\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
\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( "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 $\"-96\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*(-96) = -96
\n" ); document.write( "2*(-48) = -96
\n" ); document.write( "3*(-32) = -96
\n" ); document.write( "4*(-24) = -96
\n" ); document.write( "6*(-16) = -96
\n" ); document.write( "8*(-12) = -96
\n" ); document.write( "(-1)*(96) = -96
\n" ); document.write( "(-2)*(48) = -96
\n" ); document.write( "(-3)*(32) = -96
\n" ); document.write( "(-4)*(24) = -96
\n" ); document.write( "(-6)*(16) = -96
\n" ); document.write( "(-8)*(12) = -96\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 $\"20\"$ :\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 the table, we can see that the two numbers $\"-4\"$ and $\"24\"$ add to $\"20\"$ (the middle coefficient).\r
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\n" ); document.write( "\n" ); document.write( "So the two numbers $\"-4\"$ and $\"24\"$ both multiply to $\"-96\"$ and add to $\"20\"$ \r
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\n" ); document.write( "\n" ); document.write( "Now replace the middle term $\"20s\"$ with $\"-4s%2B24s\"$ . Remember, $\"-4\"$ and $\"24\"$ add to $\"20\"$ . So this shows us that $\"-4s%2B24s=20s\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"3s%5E2%2Bhighlight%28-4s%2B24s%29-32\"$ Replace the second term $\"20s\"$ with $\"-4s%2B24s\"$ .\r
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\n" ); document.write( "\n" ); document.write( " $\"%283s%5E2-4s%29%2B%2824s-32%29\"$ Group the terms into two pairs.\r
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\n" ); document.write( "\n" ); document.write( " $\"s%283s-4%29%2B%2824s-32%29\"$ Factor out the GCF $\"s\"$ from the first group.\r
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\n" ); document.write( "\n" ); document.write( " $\"s%283s-4%29%2B8%283s-4%29\"$ Factor out $\"8\"$ 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( " $\"%28s%2B8%29%283s-4%29\"$ Combine like terms. Or factor out the common term $\"3s-4\"$ \r
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\n" ); document.write( "\n" ); document.write( "So $\"2%283s%5E2%2B20s-32%29\"$ then factors further to $\"2%28s%2B8%29%283s-4%29\"$ \r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"6s%5E2%2B40s-64\"$ completely factors to $\"2%28s%2B8%29%283s-4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "In other words, $\"6s%5E2%2B40s-64=2%28s%2B8%29%283s-4%29\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Note: you can check the answer by expanding $\"2%28s%2B8%29%283s-4%29\"$ to get $\"6s%5E2%2B40s-64\"$ or by graphing the original expression and the answer (the two graphs should be identical).\r
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\n" ); document.write( "\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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\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

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