It is given that the polynomial $P(x) = x^3 + ax^2 + bx + c$ has three distinct positive integer roots and $P(22) = 21$. Let $Q(x) = x^2 - 2x + 22$. It is also given that $P(Q(x))$ has no real roots then $a$ is equal to:
Step-by-Step Solution
Key Concept: Finding the range of a composite function and using product factorization determines the roots and coefficients.
Note that $Q(x) = (x-1)^2 + 21$. Since $Q(x)$ takes all values in $[21, \infty)$ and $P(Q(x))$ has no real roots, all three roots of $P(x)$ must be less than 21. From $P(22) = 21$, we get $(22-x_1)(22-x_2)(22-x_3) = 1 \times 3 \times 7$ where $x_1, x_2, x_3$ are roots. Thus $x_1 = 15, x_2 = 19, x_3 = 21$, giving $a = -(x_1+x_2+x_3) = -55$.
Correct Answer: 2