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

Concept: If a polynomial of degree 3 is divisible by a polynomial of degree 2, then the quotient must be a linear polynomial. Coefficient comparison is used to equate corresponding terms and apply the given constraints.
Step 1: Assume the divisibility condition Let: \[ x^3 + ax^2 + bx + c = (x^2 + 2)(x + p) \] where \( p \in \mathbb{N} \).
Step 2: Expand the right-hand side \[ (x^2 + 2)(x + p) = x^3 + px^2 + 2x + 2p \]
Step 3: Compare coefficients Comparing with \( x^3 + ax^2 + bx + c \), we get: \[ a = p,\quad b = 2,\quad c = 2p \]
Step 4: Apply the given constraints Given: \[ a, b, c \le 20 \] From \( b = 2 \), the condition is satisfied. From \( c = 2p \le 20 \Rightarrow p \le 10 \) Also, since \( a = p \in \mathbb{N} \), possible values of \( p \) are: \[ p = 1, 2, 3, \ldots, 10 \]
Step 5: Count the number of polynomials Each value of \( p \) gives one distinct polynomial. \[ \text{Total number of polynomials} = 10 \]