SOLUTION: The equation {{{x^4-5x^3-9x^2+ax+b=0}}} has a triple root. Given this root is an integer find the triple root and the value of b.

It is well-known that for the 4th degree polynomial:



The sum of its roots taken one at a time = -B/A

The sum of all products of its roots taken two at a time = C/A

The sum of all products of its roots taken three at a time = -D/A

The sum of all products of its roots taken four at a time = E/A

In  we are told that one root is a
triple root.  

So let the four roots be r, r, r, and s

Then A=1, B=-5, C=-9, D=a, E=b

The sum of roots taken one at a time = 
r+r+r+s = -(-5)/1 or 3r+s = 5

The sum of all products of its roots taken two at a time = 
rr+rr+rs+rr+rs+rs = -9/1 or 3r²+3rs = -9 or r²+rs = -3

So we have this system of equations:

 

Solve the first equation for s, s = 5-3r

Substituting in the second equation,






r-3 = 0;  2r+1 = 0
  r = 3     2r = -1
             r = -1/2

We are told that the triple root is an integer, so we
discard the -1/2, and take r = 3

Substituting r = 3 in 

s = 5-3r
s = 5-3(3)
s = 5-9
s = -4

The sum of all products of its roots taken three at a time = -a/1, so
rrr+rrs+rrs+rrs = -a or r³+3r²s = -a, substituting for r and s:

     r³+3r²s = -a 
3³+3(3)²(-4) = -a
 27+3(9)(-4) = -a
      27-109 = -a
         -81 = -a
           a = 81

The sum of all products of its roots taken four at a time = b/1
That's all of its roots, and there is but one product, so
rrrs = b, or

   r³s = b
3³(-4) = b
27(-4) = b
  -108 = b

Edwin