SOLUTION: Sorry if this is the wrong topic but Is this even possible to factor out a binomial? 6x^2+17x-40

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.

===============================================================

Answer:

So doesn't factor at all (over the rational numbers).

So is prime.

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