To determine the number of polynomials in the set \( S = \{x^3 + ax^2 + bx + c : a, b, c \in \mathbb{N} \text{ and } a, b, c \leq 20\} \) that are divisible by \( x^2 + 2 \), we'll follow these steps:
Step 1: Understand the Divisibility Condition
A polynomial \( f(x) \) is divisible by \( x^2 + 2 \) if and only if the remainder when \( f(x) \) is divided by \( x^2 + 2 \) is zero. This means:
f(x) = (x^2 + 2)q(x) \text{ for some polynomial } q(x).
The polynomial \( x^2 + 2 \) is of degree 2, so \( f(x) \) can be expressed as \( (x^2 + 2)q(x) = x^3 + ax^2 + bx + c \), where \( q(x) \) is a linear polynomial of degree 1, \( q(x) = dx + e \). Expanding this, we have:
(x^2 + 2)(dx + e) = dx^3 + ex^2 + 2dx + 2e.
For these polynomials to match, we compare the coefficients:
- Coefficient of \( x^3 \): \( d = 1 \) (since the highest degree term in our polynomial is \( x^3 \))
- Coefficient of \( x^2 \): \( e = a \)
- Coefficient of \( x \): \( 2d = b \Rightarrow b = 2 \cdot 1 = 2 \)
- Constant term: \( 2e = c \Rightarrow c = 2a \)
Step 2: Set the Parameter Constraints
We have determined the relations: \( b = 2 \) and \( c = 2a \). Given the constraints \( a, b, c \leq 20 \), we find the allowable values for \( a \):
c = 2a \leq 20 \Rightarrow a \leq 10.
Step 3: Count the Number of Possible Polynomials
The possible values for \( a \) are all natural numbers from 1 to 10. For each \( a \), there is exactly one polynomial because once \( a \) is chosen, \( b \) and \( c \) are uniquely determined. Therefore, there are 10 polynomials:
- For \( a = 1 \), \( b = 2 \), \( c = 2 \times 1 = 2 \)
- For \( a = 2 \), \( b = 2 \), \( c = 2 \times 2 = 4 \)
- ...
- For \( a = 10 \), \( b = 2 \), \( c = 2 \times 10 = 20 \)
Conclusion:
The number of such polynomials is 10.
Thus, the correct answer is 10.
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