Q28 / 75 · JEE Main 2026 (28 January Shift 1) · Mathematics
Let $\mathrm{S}=\left\{x^{3}+a x^{2}+b x+c: a, b, c \in \mathrm{~N}\right.$ and $\left.a, b, c \leq 20\right\}$ be a set of polynomials. Then the number of polynomials in S, which are divisible by $x^{2}+2$, is
- Correct answerA10
- B20
- C6
- D120
Solution
If $x^2 + 2$ divides $x^3 + ax^2 + bx + c$, write $x^3 + ax^2 + bx + c = (x^2 + 2)(x + k)$.
Expanding: $x^3 + kx^2 + 2x + 2k$.
Comparing coefficients: $a = k$, $b = 2$, $c = 2a$.
Constraints: $a \in \mathbb{N}$, $a \leq 20$, and $c = 2a \leq 20 \Rightarrow a \leq 10$.
So $a \in \{1, 2, 3, ..., 10\}$, giving 10 polynomials.
Expanding: $x^3 + kx^2 + 2x + 2k$.
Comparing coefficients: $a = k$, $b = 2$, $c = 2a$.
Constraints: $a \in \mathbb{N}$, $a \leq 20$, and $c = 2a \leq 20 \Rightarrow a \leq 10$.
So $a \in \{1, 2, 3, ..., 10\}$, giving 10 polynomials.
Hence, the correct answer is option: 'A'.