Polynomial identity or factoring to simplify a given expression

The question requires using a polynomial identity (e.g., sum of cubes, difference of squares) or factoring a polynomial expression given an algebraic constraint to arrive at a simplified result.

jee-main 2025 Q6 View
The product of all the rational roots of the equation $\left(x^2 - 9x + 11\right)^2 - (x-4)(x-5) = 3$, is equal to
(1) 14
(2) 21
(3) 28
(4) 7
turkey-yks 2011 Q6 View
Let x and y be real numbers such that
$$\begin{aligned} & x ^ { 3 } - 3 x ^ { 2 } y = 3 \\ & y ^ { 3 } - 3 x y ^ { 2 } = 11 \end{aligned}$$
Accordingly, what is the difference $x - y$?
A) 3
B) 2
C) 1
D) - 2
E) - 3