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 :
1,2,3,5,6,10,15,30
-1,-2,-3,-5,-6,-10,-15,-30
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*30 = 30
2*15 = 30
3*10 = 30
5*6 = 30
(-1)*(-30) = 30
(-2)*(-15) = 30
(-3)*(-10) = 30
(-5)*(-6) = 30
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 | 30 | 1+30=31 | | 2 | 15 | 2+15=17 | | 3 | 10 | 3+10=13 | | 5 | 6 | 5+6=11 | | -1 | -30 | -1+(-30)=-31 | | -2 | -15 | -2+(-15)=-17 | | -3 | -10 | -3+(-10)=-13 | | -5 | -6 | -5+(-6)=-11 |
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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