Question 577749: Factor; if cant, write prime: 64p^2 - 63p + 16 (no equal sign by the way) just factor
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
Looking at the expression  , we can see that the first coefficient is  , the second coefficient is  , and the last term is  .
Now multiply the first coefficient by the last term to get  .
Now the question is: what two whole numbers multiply to  (the previous product) and add to the second coefficient ?
To find these two numbers, we need to list all of the factors of (the previous product).
Factors of :
1,2,4,8,16,32,64,128,256,512,1024
-1,-2,-4,-8,-16,-32,-64,-128,-256,-512,-1024
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*1024 = 1024
2*512 = 1024
4*256 = 1024
8*128 = 1024
16*64 = 1024
32*32 = 1024
(-1)*(-1024) = 1024
(-2)*(-512) = 1024
(-4)*(-256) = 1024
(-8)*(-128) = 1024
(-16)*(-64) = 1024
(-32)*(-32) = 1024
Now let's add up each pair of factors to see if one pair adds to the middle coefficient :
| 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 |
From the table, we can see that there are no pairs of numbers which add to . So cannot be factored.
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Answer:
So doesn't factor at all (over the rational numbers).
So is prime.
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