# Understand the problem

Find all the real Polynomials P(x) such that it satisfies the functional equation: .

##### Source of the problem

Unknown

##### Topic

Functional Equation, Polynomials

##### Difficulty Level

7/10

##### Suggested Book

Excursion in Mathematics Challenges and Thrills in Pre College Mathematics

# Start with hints

Do you really need a hint? Try it first!

Well, it is really good that the information polynomial is given! You should use that. What is the first thing that you check in a Polynomial Identity? Degree! Yes, check whether the degree of the Polynomial on both the LHS and RHS are the same or not. Yes, they are both the same . But did you observe something fishy?

Now rewrite the equation as . Do the Degree trick now… You see it right? Yes, on the left it is and on the RHS it is . So, there are two cases now… Figure them out!

Case 1: … i.e. P(x) is either a quadratic or a constant function. Case 2: has coefficient zero till . We will study case 1 now. Case 1: … i.e. P(x) is either a quadratic or a constant function. = where . Now, expand using , it gives … Now find out all such polynomials satisfying this property. For e.g. is a solution. If P(x) is constant, prove that .

Case 2: . Assume a general form of P(x) = $latex $and show that P(x) must be quadratic or lesser degree by comparing coefficients as you have a quadratic on RHS and n degree polynomial of the LHS. Now, we have already solved it for quadratic or less degree.

- Always
**Compare the Degree of Polynomials**in identities like this. It provides a lot of information. - Compare the coefficients of Polynomials on both sides to equalize the coefficient on both sides.

- Find all polynomials .
- Find all polynomials \( P(cP(x)) – d P(P(x)) = P(x)^{2} \) depending on the values of c and d.

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