Question 640515: Sorry if this is the wrong topic but
Is this even possible to factor out a binomial?
6x^2+17x-40
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 :
1,2,3,4,5,6,8,10,12,15,16,20,24,30,40,48,60,80,120,240
-1,-2,-3,-4,-5,-6,-8,-10,-12,-15,-16,-20,-24,-30,-40,-48,-60,-80,-120,-240
Note: list the negative of each factor. This will allow us to find all possible combinations.
These factors pair up and multiply to .
1*(-240) = -240
2*(-120) = -240
3*(-80) = -240
4*(-60) = -240
5*(-48) = -240
6*(-40) = -240
8*(-30) = -240
10*(-24) = -240
12*(-20) = -240
15*(-16) = -240
(-1)*(240) = -240
(-2)*(120) = -240
(-3)*(80) = -240
(-4)*(60) = -240
(-5)*(48) = -240
(-6)*(40) = -240
(-8)*(30) = -240
(-10)*(24) = -240
(-12)*(20) = -240
(-15)*(16) = -240
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 | -240 | 1+(-240)=-239 | | 2 | -120 | 2+(-120)=-118 | | 3 | -80 | 3+(-80)=-77 | | 4 | -60 | 4+(-60)=-56 | | 5 | -48 | 5+(-48)=-43 | | 6 | -40 | 6+(-40)=-34 | | 8 | -30 | 8+(-30)=-22 | | 10 | -24 | 10+(-24)=-14 | | 12 | -20 | 12+(-20)=-8 | | 15 | -16 | 15+(-16)=-1 | | -1 | 240 | -1+240=239 | | -2 | 120 | -2+120=118 | | -3 | 80 | -3+80=77 | | -4 | 60 | -4+60=56 | | -5 | 48 | -5+48=43 | | -6 | 40 | -6+40=34 | | -8 | 30 | -8+30=22 | | -10 | 24 | -10+24=14 | | -12 | 20 | -12+20=8 | | -15 | 16 | -15+16=1 |
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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