We need to determine the number of polynomials of the form \(x^3 + ax^2 + bx + c\) that are divisible by \(x^2 + 1\), where \(a, b, c \in \{1, 2, 3, \dots, 10\}\).
Since \(x^2 + 1\) is a factor of the polynomial, it must divide it completely, leading to the condition:
1. The remainder when this polynomial is divided by \(x^2 + 1\) should be zero.
2. The roots of the polynomial \(x^2 + 1 = 0\) are \(x = i\) and \(x = -i\).
This means both \(i\) and \(-i\) should satisfy the polynomial equation:
\(x^3 + ax^2 + bx + c = 0\) for \(x = i\) and \(x = -i\).
Substitute \(x = i\) into the polynomial:
\(i^3 + ai^2 + bi + c = 0\)
We know \(i^2 = -1\) and \(i^3 = -i\), so the equation becomes:
\(-i - a + bi + c = 0\)
- which is equivalent to \((b - 1)i + (c - a) = 0\).
This implies that:
\(b - 1 = 0 \quad \text{and} \quad c - a = 0\)
We conclude:
Now, substituting \(x = -i\) into the polynomial:
\((-i)^3 + a(-i)^2 + b(-i) + c = 0\)
Simplify this using \((-i)^2 = -1\) and \((-i)^3 = i\):
\(i + a(-1) - bi + c = 0\)
- which becomes \((1 - b)i + (c - a) = 0\)
We notice it matches our previous conditions:
Given that \(a, b, c \in \{1, 2, 3, \dots, 10\}\), let's list the valid polynomials:
Since \(b = 1\), and \(c = a\), we can choose \(a\) in 10 different ways (from 1 to 10).
Therefore, the total number of such polynomials that satisfy these conditions is 10.