JEE Main 2026 (Online) 28th January Morning ShiftQuestion 1
Let $S=\{x^3+ax^2+bx+c:a,b,c\in\mathbf{N}\text{ and }a,b,c\le20\}$ be a set of polynomials. Then the number of polynomials in $S$ which are divisible by $x^2+2$ is :
Quadratic Equation · Higher Degree Polynomial Equations
Solution
Concept. Reduce the cubic modulo $x^2+2$ (replace $x^2$ by $-2$); divisibility means the remainder is $0$. Formulas used. (1) Modulo $x^2+2$: $x^2\equiv-2$ and $x^3\equiv-2x$Step 1 - Remainder. By (1), $$x^3+ax^2+bx+c\equiv-2x-2a+bx+c=(b-2)x+(c-2a).$$ Step 2 - Zero remainder. $b=2$ and $c=2a$. Step 3 - Count. $a,c\in\mathbb N$, $c=2a\le20$, so $a=1,2,\dots,10$: $10$ polynomials. Answer: (A)