Are the binomial coefficients unique?

Question

Let $a,x,b,y$ be integers. Can we find rationals $u,v,w,t$ such that $$(ax+by)^3=ux^3+vx^2y+wxy^2+ty^3\neq 0$$ where $$(u,v,w,t)\neq ( 1, 3a^2b, 3ab^2, 1)$$ The answer looks trivial but can one prove it?

Answer 1

$$ux^3+vx^2y+wxy^2+ty^3=a^3x^3+3a^2byx^2+3ab^2y^2x+b^3y^3$$ is an equaliy involving two polynomials respect to x. since $ ( x^3, x^2, x, 1)$ is a basis then, $$(u,v,w,t)= ( a^3, 3a^2b, 3ab^2, b^3)$$