Question

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:

• A. \(120\)
• B. \(10\)
• C. \(20\)
• D. \(6\)

Hint:

When checking divisibility of polynomials, always equate coefficients after expansion and then apply the given bounds carefully.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
If a cubic polynomial \(P(x)\) is divisible by a quadratic \(Q(x)\), then \(P(x) = Q(x)(x + k)\) for some constant \(k\).
Step 2: Detailed Explanation:
Let \(x^3 + ax^2 + bx + c = (x^2 + 2)(x + k)\).
Expanding the right side:
\(x^3 + kx^2 + 2x + 2k\).
By comparing coefficients:
\(a = k\)
\(b = 2\)
\(c = 2k\)
Given \(a, b, c \in \mathbb{N}\) and \(a, b, c \leq 20\).
1. \(b = 2\) is a natural number and \(2 \leq 20\). (Satisfied)
2. \(a = k \in \mathbb{N}\).
3. \(c = 2k \in \mathbb{N}\).
Constraints:
\(k \leq 20\) and \(2k \leq 20 \Rightarrow k \leq 10\).
Since \(k\) must be a natural number (because \(a, c\) are natural numbers), the possible values for \(k\) are \(\{1, 2, 3, \dots, 10\}\).
There are 10 such values of \(k\), and each corresponds to exactly one set of coefficients \((a, b, c)\).
Step 3: Final Answer:
The number of such polynomials is 10.