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 :
1,2,4,5,10,20
-1,-2,-4,-5,-10,-20
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*20 = 20
2*10 = 20
4*5 = 20
(-1)*(-20) = 20
(-2)*(-10) = 20
(-4)*(-5) = 20
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 | 20 | 1+20=21 | | 2 | 10 | 2+10=12 | | 4 | 5 | 4+5=9 | | -1 | -20 | -1+(-20)=-21 | | -2 | -10 | -2+(-10)=-12 | | -4 | -5 | -4+(-5)=-9 |
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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