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 :
1,2,3,4,6,12,23,46,69,92,138,276
-1,-2,-3,-4,-6,-12,-23,-46,-69,-92,-138,-276
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*(-276) = -276
2*(-138) = -276
3*(-92) = -276
4*(-69) = -276
6*(-46) = -276
12*(-23) = -276
(-1)*(276) = -276
(-2)*(138) = -276
(-3)*(92) = -276
(-4)*(69) = -276
(-6)*(46) = -276
(-12)*(23) = -276
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 | -276 | 1+(-276)=-275 | | 2 | -138 | 2+(-138)=-136 | | 3 | -92 | 3+(-92)=-89 | | 4 | -69 | 4+(-69)=-65 | | 6 | -46 | 6+(-46)=-40 | | 12 | -23 | 12+(-23)=-11 | | -1 | 276 | -1+276=275 | | -2 | 138 | -2+138=136 | | -3 | 92 | -3+92=89 | | -4 | 69 | -4+69=65 | | -6 | 46 | -6+46=40 | | -12 | 23 | -12+23=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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