Question 681794: Help explain why this expression cannot be factored
40a^2+21a-2
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 :
1,2,4,5,8,10,16,20,40,80
-1,-2,-4,-5,-8,-10,-16,-20,-40,-80
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*(-80) = -80
2*(-40) = -80
4*(-20) = -80
5*(-16) = -80
8*(-10) = -80
(-1)*(80) = -80
(-2)*(40) = -80
(-4)*(20) = -80
(-5)*(16) = -80
(-8)*(10) = -80
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 | -80 | 1+(-80)=-79 | | 2 | -40 | 2+(-40)=-38 | | 4 | -20 | 4+(-20)=-16 | | 5 | -16 | 5+(-16)=-11 | | 8 | -10 | 8+(-10)=-2 | | -1 | 80 | -1+80=79 | | -2 | 40 | -2+40=38 | | -4 | 20 | -4+20=16 | | -5 | 16 | -5+16=11 | | -8 | 10 | -8+10=2 |
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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