document.write( "Question 385276: Not understanding, can anyone help?\r
\n" ); document.write( "\n" ); document.write( "Factor the expression a^2-6ab+96b^2 into a product of binomials.\r
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Algebra.Com's Answer #272583 by jim_thompson5910(35256) $\"\"$ $\"About$

You can put this solution on YOUR website!
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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"a%5E2-6ab%2B96b%5E2\"$ , we can see that the first coefficient is $\"1\"$ , the second coefficient is $\"-6\"$ , and the last coefficient is $\"96\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"1\"$ by the last coefficient $\"96\"$ to get $\"%281%29%2896%29=96\"$ .\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 $\"-6\"$ ?\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 $\"-6\"$ :\r
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First Number	Second Number	Sum
1	96	1+96=97
2	48	2+48=50
3	32	3+32=35
4	24	4+24=28
6	16	6+16=22
8	12	8+12=20
-1	-96	-1+(-96)=-97
-2	-48	-2+(-48)=-50
-3	-32	-3+(-32)=-35
-4	-24	-4+(-24)=-28
-6	-16	-6+(-16)=-22
-8	-12	-8+(-12)=-20

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\n" ); document.write( "\n" ); document.write( "From the table, we can see that there are no pairs of numbers which add to $\"-6\"$ . So $\"a%5E2-6ab%2B96b%5E2\"$ cannot be factored.\r
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\n" ); document.write( "\n" ); document.write( "Answer:\r
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\n" ); document.write( "\n" ); document.write( "So $\"a%5E2-6ab%2B96b%5E2\"$ doesn't factor at all (over the rational numbers).\r
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\n" ); document.write( "\n" ); document.write( "So $\"a%5E2-6ab%2B96b%5E2\"$ is prime.\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, feel free to check out my tutoring website\r
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\n" ); document.write( "\n" ); document.write( "Jim \n" ); document.write( "

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