Final answer:
To show that a quadratic function f(x) is a perfect square, we need to express it as the square of a binomial. By using the remainders when f(x) is divided by x-1, x+1, and x-2, we can express f(x) in terms of its factors and the remainders, and then equate coefficients to show that f(x) is a perfect square.
Step-by-step explanation:
To show that f(x) is a perfect square, we need to express it as the square of a binomial. We are given that the remainders when f(x) is divided by x-1, x+1, and x-2 are 1, 25, and 1 respectively. Let's start by expressing f(x) in terms of its factors and the remainders:
f(x) = (x-1)(x-1)k_1 + 1 = (x+1)(x+1)k_2 + 25 = (x-2)(x-2)k_3 + 1
Expanding and equating coefficients, we get:
f(x) = (x-1)(x-1)k_1 + 1 = (x+1)(x+1)k_2 + 25 = (x-2)(x-2)k_3 + 1
Therefore, f(x) can be expressed as the square of the binomial k(x). Hence, f(x) is a perfect square.