How to Find All Polynomials Satisfying Specific Conditions

When faced with the task of finding all polynomials with real coefficients that meet certain conditions, such as p_0 0 and p_{x 1}^3 p_x^3 , we need to approach the problem step-by-step. Let's delve into the detailed analysis and steps involved in solving this problem.

Initial Conditions

Given that p_0 0 , we can express the polynomial p_x as:

p_x x q_x , where q_x is some polynomial.

This expression simplifies our problem by allowing us to focus on the polynomial q_x rather than p_x .

Substituting into the Functional Equation

Substituting p_x into the equation p_{x 1}^3 p_x^3 , we get:

p_{x 1}^3 (x 1)^3 q_{x 1}^3 and p_x^3 x^3 q_x^3 .

Expanding the Right-Hand Side

Expanding (x 1)^3 q_{x 1}^3 using the binomial theorem, we obtain:

(x 1)^3 q_{x 1}^3 (x^3 3x^2 3x 1) q_x^3 .

Setting Up the Equation

This gives us:

(x 1)^3 q_{x 1}^3 x^3 q_x^3 - 3x^2 q_x^2 - 3x q_x - 1 .

Analyzing the Degrees

The left side (x 1)^3 q_{x 1}^3 has a degree that is determined by the degree of q_x . If q_x is a polynomial of degree n , then p_x is a polynomial of degree n-1 . The left-hand side has degree 3 - 3n , and the right-hand side has degree 3 - 1 , suggesting that the degrees must match.

Matching Degrees

From the degrees, we can conclude:

3 - 3n 3 - 1 , which simplifies to:

3n 1 , implying n frac{1}{3} .

Since n must be a non-negative integer, the only solution is n 0 , meaning that q_x is a constant polynomial.

Finding q_x

Let q_x c , where c is a constant. Thus:

p_x c x .

Substituting p_x cx back into the original equation:

(cx 1)^3 (cx)^3 and (cx-1)^3 (cx)^3 - 3(cx)^2 - 3(cx) - 1 .

Equating the Two Expressions

This equality must hold for all x . Expanding both sides gives:

c^3 x^3 3c^2 x^2 3cx 1 c^3 x^3 - 3c^2 x^2 - 3cx - 1 .

Comparing Coefficients

We can compare coefficients:

For x^3 term: c c^3

From c c^3 , we get c(c^2 - 1) 0 , which implies:

c 0 or c 1 .

For x^2 term: 3c -3c^2

From 3c -3c^2 , we get c c^2 0 , which also implies:

c 0 or c 1 .

For x term: 3c -3c , which is trivially true. For constant term: c -1 , which contradicts our earlier conditions.

Final Solutions

Hence, the possible values for c are:

If c 0 , then p_x 0 . If c 1 , then p_x x .

Thus, the polynomials that satisfy the given conditions are:

boxed{p_x 0 text{ or } p_x x.}