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 :
1,2,4,13,26,52
-1,-2,-4,-13,-26,-52
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*52 = 52
2*26 = 52
4*13 = 52
(-1)*(-52) = 52
(-2)*(-26) = 52
(-4)*(-13) = 52
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 | 52 | 1+52=53 | | 2 | 26 | 2+26=28 | | 4 | 13 | 4+13=17 | | -1 | -52 | -1+(-52)=-53 | | -2 | -26 | -2+(-26)=-28 | | -4 | -13 | -4+(-13)=-17 |
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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