The Rational Root Theorem states that any potential rational root of a polynomial function can be expressed as a fraction, where the numerator is a factor of the constant term and the denominator is a factor of the leading coefficient.
To determine if 25 could be a potential rational root, we first identify a polynomial where 25 is included in the list of possible rational roots. A simple polynomial to consider could be:
f(x) = x^2 – 25
In this case, the constant term is -25, and the leading coefficient is 1. The factors of -25 (the constant term) are ±1, ±5, ±25. The factors of 1 (the leading coefficient) are just ±1. By applying the Rational Root Theorem, the potential rational roots for this polynomial are ±1, ±5, and ±25.
Since 25 is a factor of -25 and the only leading coefficient factor being ±1, we can conclude that 25 is indeed a potential rational root of the polynomial f(x) = x^2 – 25. Furthermore, if we substitute 25 into the polynomial:
f(25) = 25^2 – 25 = 625 – 25 = 600,
we see that 25 is not an actual root because it does not result in zero. However, it remains a potential rational root according to the theorem.