Polynomials

Step 1: Finding roots of the equation. Use the quadratic or factorization method to find the roots of the quartic equation. The non-integer roots are found to be .
Step 2: Conclusion. Thus, the non-integer roots of the equation are .

Step 1: Factor divisor.

Step 2: Use factor theorem.
If divisible, then:

Step 3: Apply .


Step 4: Apply .



Step 5: Solve simultaneous equations.


Subtract:

Then:

Final Answer:

Step 1: Determine the form of . The given functional equation is: Rearranging, we get: This suggests that is of the form , where is an integer. Let:
Step 2: Use the given condition . Thus, .
Step 3: Compute the derivative .
Step 4: Analyze . Given are in GP, let: where is the common ratio. Then: Thus, the sequence is: which is a geometric progression (GP).
Step 5: Determine the correct option. The sequence is in GP.