The number of monic quadratic polynomials of the form $x^2 + ax + b$ with integer roots, where $1, a, b$ are in AP is(are):
Step-by-Step Solution
Key Concept: Convert the AP condition into a Diophantine equation and factor it as $(p+2)(q+2) = 3$ to find all integer root pairs.
Let the integer roots be $p$ and $q$. Then $x^2 + ax + b = (x-p)(x-q)$, giving us $a = -(p+q)$ and $b = pq$. Since $1, a, b$ are in AP, we have $2a = 1 + b$, or $b = 2a - 1$. Substituting: $pq = 2(-(p+q)) - 1$, which simplifies to $pq + 2p + 2q + 1 = 0$. Factoring: $(p+2)(q+2) = 3$. Since $p, q$ are integers, $(p+2, q+2)$ must be factor pairs of 3: $(1,3), (3,1), (-1,-3), (-3,-1)$. This gives $(p,q) = (-1,1), (1,-1), (-3,-5), (-5,-3)$. The distinct polynomials are $x^2 - 1$ and $x^2 - 8x + 15$, yielding exactly 2 distinct monic quadratic polynomials.
Correct Answer: 3