f(x) and g(x) are monic quadratic polynomials that satisfy the following conditions:
f(x)=0 has real distinct roots a1 and a2.
g(x)=0 has real distinct roots b1 and b2.
{f(b1),f(b2)}={b1,b2}. (set of f(b1),f(b2) is same as b1, b2)
{g(a1),g(a2)}={a1,a2}. (set of f(a1),f(a2) is same as a1, a2)
f(1)g(1)=132.
What is the value of f(2)g(2)?
To find the value of f(2)g(2), we need to first find the values of f(1) and g(1).
Given that f(x) is a monic quadratic polynomial with real distinct roots a1 and a2, we can express f(x) as:
f(x) = (x - a1)(x - a2)
Given that g(x) is a monic quadratic polynomial with real distinct roots b1 and b2, we can express g(x) as:
g(x) = (x - b1)(x - b2)
Using the conditions given, we have {f(b1), f(b2)} = {b1, b2}. This means that plugging b1 and b2 into f(x) should give us b1 and b2. We can write this condition as:
f(b1) = b1 and f(b2) = b2
Similarly, using the condition {g(a1), g(a2)} = {a1, a2}, we can write:
g(a1) = a1 and g(a2) = a2
Now, we are given that f(1)g(1) = 132. Plugging in the expressions for f(x) and g(x), we have:
[(1 - a1)(1 - a2)][(1 - b1)(1 - b2)] = 132
Expanding this expression, we get:
(1 - a1)(1 - a2)(1 - b1)(1 - b2) = 132
Given this equation, we can solve for f(1) and g(1). Once we find these values, we can find f(2)g(2) by evaluating the expressions f(2) and g(2) in a similar manner.
To find the values of a1, a2, b1, and b2 and solve the equation, we might need additional information or constraints.