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Let $ S = { x^3 + ax^2 + bx + c : a, b, c \in \mathbb{N} \text{ and } a, b, c \le 20 } $ be a set of polynomials. Then the number of polynomials in $ S $, which are divisible by $ x^2 + 2 $, is
1
2
3
4
Solution
$ x^3 + ax^2 + bx + c = (x^2 + 2)\left(x + \frac{c}{2}\right) $
$ \Rightarrow a = \frac{c}{2},; b = 2 $
$ \Rightarrow b = 2,; a = \frac{c}{2} \in {2, 4, \ldots, 20} $
Number of polynomials in $ S $ will be $ 10 $
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