Final answer:
-2/5 is a potential rational root of function d) f(x) = 25x⁴ - 7x² + x + 4, according to the Rational Root Theorem, because the constant term and leading coefficient meet the necessary conditions.
Step-by-step explanation:
The Rational Root Theorem states that for a polynomial equation with integer coefficients ax^n + bx^(n-1) + … + k = 0, any rational root p/q, where p is a factor of the constant term k and q is a factor of the leading coefficient a, must be in its simplest form. In the case of -2/5 being a potential root, the constant term of the polynomial function must have 2 as a factor, and the leading coefficient must have 5 as a factor.
By examining the given options:
- a) f(x) = 4x⁴ - 7x² + x + 25 - The constant term is 25, which is not a multiple of 2.
- b) f(x) = 9x⁴ - 7x² + x + 10 - The constant term is 10, which is a multiple of 2, but the leading coefficient is 9, not a multiple of 5.
- c) f(x) = 10x⁴ - 7x² + x + 9 - The constant term is 9, which is not a multiple of 2.
- d) f(x) = 25x⁴ - 7x² + x + 4 - The constant term is 4, a multiple of 2, and the leading coefficient is 25, a multiple of 5.
Therefore, according to the Rational Root Theorem, -2/5 is a potential rational root of the function d) f(x) = 25x⁴ - 7x² + x + 4.