Let S = {x³ + ax² + bx + c : a, b, c ∈ ℕ and a, b, c ≤ 20} be a set of polynomials. Then the number of polynomials in S which are divisible by x² + 2 is ________.

Options:

1. 20

2. 6

3. 10

4. 120

Correct Answer: 10

Explanation

Step 1:

If a polynomial is divisible by x² + 2, then x = √2 i and x = −√2 i must be its roots.

Step 2:

Substitute x = √2 i in x³ + ax² + bx + c = 0

⇒ −2√2 i − 2a + b√2 i + c = 0

Step 3:

Equating real and imaginary parts:

• −2a + c = 0 ⇒ c = 2a
• −2 + b = 0 ⇒ b = 2

Step 4:

Since c ≤ 20 ⇒ 2a ≤ 20 ⇒ a ≤ 10 So, a can take 10 values.

Hence, total number of polynomials = 10

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