Question

Find the value of a and B so that the polynomial x cube minus 10 X square + A x+ B is exactly divisible by 8 minus 1 as well as x minus 2

Answer 1

The arithmetic mean of the roots is 8/5. 

So we can think of the roots as: 
r = 8/5 - d 
s = 8/5 + d 

Their reciprocals are: 
1/r = 1/(8/5 - d) 
1/s = 1/(8/5 + d) 

The arithmetic mean of those two numbers is: 
(1/r + 1/s) * 1/2 = [ (8/5 + d) + (8/5 - d) ] / 2[ (8/5 - d)(8/5 + d) ] 

= (16/5) / 2((8/5)² - d²) 
(8/5) / ((8/5)² - d²) = 8/7 

Cross multiply: 
8*((8/5)² - d²) = 7*8/5 
(8/5)² - d² = 7/5 

d² = (8/5)² - 7/5 
d² = 64/25 - 35/25) 
d² = 29/25 
d = ±√29/5 

So we have: 
x = (8 ± √29) / 5 

2a = 5 
a = 5/2 
-b = 8 

b² - 4ac = 29 
64 - 4(5/2)c = 29 
64 - 10c = 29 
35 = 10c 
c = 7/2 

Answer: 
f(x) = (5/2)x² - 8x + 7/2 
or 
f(x) = ½(5x² - 16x + 7) 

We can also multiply that by 2 to get the same roots: 
f(x) = 5x² - 16x + 7