document.write( "Question 577749: Factor; if cant, write prime: 64p^2 - 63p + 16 (no equal sign by the way) just factor \n" ); document.write( "

Algebra.Com's Answer #370280 by jim_thompson5910(35256) $\"\"$ $\"About$

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\n" ); document.write( "\n" ); document.write( "Looking at the expression $\"64p%5E2-63p%2B16\"$ , we can see that the first coefficient is $\"64\"$ , the second coefficient is $\"-63\"$ , and the last term is $\"16\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now multiply the first coefficient $\"64\"$ by the last term $\"16\"$ to get $\"%2864%29%2816%29=1024\"$ .\r
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\n" ); document.write( "\n" ); document.write( "Now the question is: what two whole numbers multiply to $\"1024\"$ (the previous product) and add to the second coefficient $\"-63\"$ ?\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 $\"1024\"$ (the previous product).\r
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\n" ); document.write( "\n" ); document.write( "Factors of $\"1024\"$ :\r
\n" ); document.write( "\n" ); document.write( "1,2,4,8,16,32,64,128,256,512,1024\r
\n" ); document.write( "\n" ); document.write( "-1,-2,-4,-8,-16,-32,-64,-128,-256,-512,-1024\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 $\"1024\"$ .\r
\n" ); document.write( "\n" ); document.write( "1*1024 = 1024
\n" ); document.write( "2*512 = 1024
\n" ); document.write( "4*256 = 1024
\n" ); document.write( "8*128 = 1024
\n" ); document.write( "16*64 = 1024
\n" ); document.write( "32*32 = 1024
\n" ); document.write( "(-1)*(-1024) = 1024
\n" ); document.write( "(-2)*(-512) = 1024
\n" ); document.write( "(-4)*(-256) = 1024
\n" ); document.write( "(-8)*(-128) = 1024
\n" ); document.write( "(-16)*(-64) = 1024
\n" ); document.write( "(-32)*(-32) = 1024\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 $\"-63\"$ :\r
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First Number	Second Number	Sum
1	1024	1+1024=1025
2	512	2+512=514
4	256	4+256=260
8	128	8+128=136
16	64	16+64=80
32	32	32+32=64
-1	-1024	-1+(-1024)=-1025
-2	-512	-2+(-512)=-514
-4	-256	-4+(-256)=-260
-8	-128	-8+(-128)=-136
-16	-64	-16+(-64)=-80
-32	-32	-32+(-32)=-64

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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 $\"-63\"$ . So $\"64p%5E2-63p%2B16\"$ 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 $\"64p%5E2-63p%2B16\"$ doesn't factor at all (over the rational numbers).\r
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\n" ); document.write( "\n" ); document.write( "So $\"64p%5E2-63p%2B16\"$ is prime.
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