Question 335490: Is 7x^2 - 12x + 6 prime or can it be factored?
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website!
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 :
1,2,3,6,7,14,21,42
-1,-2,-3,-6,-7,-14,-21,-42
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*42 = 42
2*21 = 42
3*14 = 42
6*7 = 42
(-1)*(-42) = 42
(-2)*(-21) = 42
(-3)*(-14) = 42
(-6)*(-7) = 42
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 | 42 | 1+42=43 | | 2 | 21 | 2+21=23 | | 3 | 14 | 3+14=17 | | 6 | 7 | 6+7=13 | | -1 | -42 | -1+(-42)=-43 | | -2 | -21 | -2+(-21)=-23 | | -3 | -14 | -3+(-14)=-17 | | -6 | -7 | -6+(-7)=-13 |
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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