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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
:
1,2,5,10,37,74,185,370
-1,-2,-5,-10,-37,-74,-185,-370
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to
.
1*(-370) = -370
2*(-185) = -370
5*(-74) = -370
10*(-37) = -370
(-1)*(370) = -370
(-2)*(185) = -370
(-5)*(74) = -370
(-10)*(37) = -370
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 | -370 | 1+(-370)=-369 |
| 2 | -185 | 2+(-185)=-183 |
| 5 | -74 | 5+(-74)=-69 |
| 10 | -37 | 10+(-37)=-27 |
| -1 | 370 | -1+370=369 |
| -2 | 185 | -2+185=183 |
| -5 | 74 | -5+74=69 |
| -10 | 37 | -10+37=27 |
From the table, we can see that there are no pairs of numbers which add to
. So
cannot be factored.
===============================================================
Answer:
So doesn't factor at all (over the rational numbers).
So is prime.
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