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 :
1,2,3,4,6,12
-1,-2,-3,-4,-6,-12
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*12 = 12
2*6 = 12
3*4 = 12
(-1)*(-12) = 12
(-2)*(-6) = 12
(-3)*(-4) = 12
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 | 12 | 1+12=13 | | 2 | 6 | 2+6=8 | | 3 | 4 | 3+4=7 | | -1 | -12 | -1+(-12)=-13 | | -2 | -6 | -2+(-6)=-8 | | -3 | -4 | -3+(-4)=-7 |
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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