Question 5
The number of polynomials of the form $$(x^3 + ax^2 + bx + c)$$ which are divisible by $$x^2 + 1$$ where $$a, b, c \in 1, 2, 3, 4, \ldots, 12$$ is
Solution
If $$x^2 + 1$$ divides the cubic, the other factor must be $$x + a$$, so $$x^3 + ax^2 + bx + c = (x^2 + 1)(x + a) = x^3 + ax^2 + x + a$$. Comparing coefficients gives $$b = 1$$ and $$c = a$$. Since $$a$$ can be any of the 12 allowed values and then $$c$$ is fixed, there are 12 such polynomials.
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