SOLUTION: Please help me factor {{{ 9x^2+89x+40 }}} I know that the first number in the bracket will probably have to be 9 or 3 but I'm not sure about how about the other numbers since 89 a

Question 538903: Please help me factor
I know that the first number in the bracket will probably have to be 9 or 3 but I'm not sure about how about the other numbers since 89 and 40 do not have a common multiple.
Thanks!
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 :
1,2,3,4,5,6,8,9,10,12,15,18,20,24,30,36,40,45,60,72,90,120,180,360
-1,-2,-3,-4,-5,-6,-8,-9,-10,-12,-15,-18,-20,-24,-30,-36,-40,-45,-60,-72,-90,-120,-180,-360

Note: list the negative of each factor. This will allow us to find all possible combinations.

These factors pair up and multiply to .
1*360 = 360
2*180 = 360
3*120 = 360
4*90 = 360
5*72 = 360
6*60 = 360
8*45 = 360
9*40 = 360
10*36 = 360
12*30 = 360
15*24 = 360
18*20 = 360
(-1)*(-360) = 360
(-2)*(-180) = 360
(-3)*(-120) = 360
(-4)*(-90) = 360
(-5)*(-72) = 360
(-6)*(-60) = 360
(-8)*(-45) = 360
(-9)*(-40) = 360
(-10)*(-36) = 360
(-12)*(-30) = 360
(-15)*(-24) = 360
(-18)*(-20) = 360

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	360	1+360=361
2	180	2+180=182
3	120	3+120=123
4	90	4+90=94
5	72	5+72=77
6	60	6+60=66
8	45	8+45=53
9	40	9+40=49
10	36	10+36=46
12	30	12+30=42
15	24	15+24=39
18	20	18+20=38
-1	-360	-1+(-360)=-361
-2	-180	-2+(-180)=-182
-3	-120	-3+(-120)=-123
-4	-90	-4+(-90)=-94
-5	-72	-5+(-72)=-77
-6	-60	-6+(-60)=-66
-8	-45	-8+(-45)=-53
-9	-40	-9+(-40)=-49
-10	-36	-10+(-36)=-46
-12	-30	-12+(-30)=-42
-15	-24	-15+(-24)=-39
-18	-20	-18+(-20)=-38

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.

If you need more help, email me at jim_thompson5910@hotmail.com

Also, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Thank you

Jim

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