The Rational Root Theorem is a useful tool in algebra for identifying possible rational roots of polynomial equations. According to the theorem, if a polynomial function is of the form:
f(x) = a_n x^n + a_(n-1) x^(n-1) + … + a_1 x + a_0
where:
- a_n is the leading coefficient, and
- a_0 is the constant term.
Then the possible rational roots of the polynomial function can be expressed as:
- p/q
Where p is a factor of the constant term (a_0) and q is a factor of the leading coefficient (a_n).
In this case, if we want to determine for which polynomial function 25 could potentially be a rational root, we need to look for a polynomial where 25 could represent p. This means that:
- The constant term (a_0) must be a multiple of 25 (i.e., it can be 25, 50, 75, etc.).
- After that, we need to choose a leading coefficient (a_n) such that its factors align with the assumption that 25 is a valid candidate.
For example, consider the polynomial:
f(x) = x^3 – 25x^2 + 100x – 125
Here,:
- The constant term (a_0 = -125) has 25 as a factor (25, -25, 1, -1, 5, -5).
- The leading coefficient (a_n = 1) has factors that include 1.
Thus, according to the Rational Root Theorem, 25 is a potential rational root of the function f(x) = x^3 – 25x^2 + 100x – 125.
In summary, any polynomial function where the constant term is divisible by 25 could potentially make 25 a rational root, provided that the other conditions of the Rational Root Theorem hold true.