SOLUTION: Help explain why this expression cannot be factored 40a^2+21a-2

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Question 681794: Help explain why this expression cannot be factored
40a^2+21a-2

Answer by jim_thompson5910(29613)   (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 NumberSecond NumberSum
1-801+(-80)=-79
2-402+(-40)=-38
4-204+(-20)=-16
5-165+(-16)=-11
8-108+(-10)=-2
-180-1+80=79
-240-2+40=38
-420-4+20=16
-516-5+16=11
-810-8+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.