Question

Total number of polynomials of the form \(x^3 + ax^2 + bx + c\) that are divisible by \(x^2 + 1\), where \(a,b,c \in \{1, 2, 3, \dots, 10\}\) is equal to

• A. 90
• B. 45
• C. 5
• D. 10

Hint:

Polynomial division condition leads to relationships between coefficients.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
If \(x^2+1\) divides the cubic, then the cubic must be \((x^2+1)(x+d)\) for some \(d\).

Step 2: Detailed Explanation:
\((x^2+1)(x+d) = x^3 + dx^2 + x + d\). Comparing with \(x^3 + ax^2 + bx + c\): \(a = d\), \(b = 1\), \(c = d\). Thus \(a = c\) and \(b = 1\). Since \(a,b,c \in \{1,\dots,10\}\), \(b=1\) is fixed, and \(a = c\) can be any of the 10 numbers from 1 to 10. So 10 polynomials.

Step 3: Final Answer:
10, which corresponds to option (D).