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 :
1,2,4,7,8,14,16,28,56,112
-1,-2,-4,-7,-8,-14,-16,-28,-56,-112
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*112 = 112
2*56 = 112
4*28 = 112
7*16 = 112
8*14 = 112
(-1)*(-112) = 112
(-2)*(-56) = 112
(-4)*(-28) = 112
(-7)*(-16) = 112
(-8)*(-14) = 112
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 | 112 | 1+112=113 | | 2 | 56 | 2+56=58 | | 4 | 28 | 4+28=32 | | 7 | 16 | 7+16=23 | | 8 | 14 | 8+14=22 | | -1 | -112 | -1+(-112)=-113 | | -2 | -56 | -2+(-56)=-58 | | -4 | -28 | -4+(-28)=-32 | | -7 | -16 | -7+(-16)=-23 | | -8 | -14 | -8+(-14)=-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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