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 :
1,3,5,7,15,21,35,105
-1,-3,-5,-7,-15,-21,-35,-105
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*105 = 105
3*35 = 105
5*21 = 105
7*15 = 105
(-1)*(-105) = 105
(-3)*(-35) = 105
(-5)*(-21) = 105
(-7)*(-15) = 105
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 | 105 | 1+105=106 | | 3 | 35 | 3+35=38 | | 5 | 21 | 5+21=26 | | 7 | 15 | 7+15=22 | | -1 | -105 | -1+(-105)=-106 | | -3 | -35 | -3+(-35)=-38 | | -5 | -21 | -5+(-21)=-26 | | -7 | -15 | -7+(-15)=-22 |
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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