Question

For which values of a & b are the zeroes of q(x)=x to power 3+2x power 2+a also the zeroes of the polymomials,p(x)=x power 5-x power 4-4x power 3+3x power 2+3x+b? Which zeroes of p(x) are not the zeroes of q(x)?

Answer 1

Divide p(x) = x⁵ - x⁴ - 4 x³ + 3x² + 3x + b 
        by  q(x) = x³ + 2x² + a
 If the zeroes of q(x) are also zeroes of p(x) then the reminder should be 0.

   x³ +2x² + a)    x⁵ - x⁴ - 4x³ + 3x² + b   ( x² - 3x + 2
                       x⁵ + 2x⁴ +ax²
                 ===============
                             -3x⁴ + (3-a)x²- 4x³
                             -3x⁴ -3ax- 6x³
                             ===================
                               2x³ +(3-a)x² + 3x(1+a) +(b-2a)
                               2x³ + 4x²  + 2a
                           ========================
                                - (a+1) x² + 3 (a+1) x +(b - 2a)

so  a = -1  and  b = +2  for the reminder to be 0.

the quotient x² - 3 x + 2 =  (x - 2) (x - 1)
   Hence the zeroes of p(x) are  1 and 2,  which are in addition to the three zeroes of q(x).

===============
we can also find the zeroes of q(x) now.
 
 q(x) = x³ + 2x² - 1
         on examining the coefficients,  x = -1  is a root.
     q(x) = (x+1) (x² + x - 1)
               x = -1  or    [-1 + √5] / 2   or   [-1 - √5]/2