When we say that x – 5 is a factor of the polynomial function f(x), we can deduce a crucial piece of information about the function. Specifically, one of the fundamentals of polynomial functions states that if a polynomial f(x) has a factor of the form x – a, then a is a root of the polynomial. In this case, since our factor is x – 5, we can conclude the following:
- f(5) = 0: This means that when we substitute x with 5 in the function f(x), the output will equal zero. Thus, 5 is a root of the function.
To clarify further, if x – 5 divides f(x) without leaving a remainder, the polynomial can be expressed in the form:
f(x) = (x - 5) * g(x)where g(x) is another polynomial. Since g(x) can be any polynomial, it could have other roots as well. However, the presence of the factor x – 5 makes it clear that 5 must definitely be a root of this polynomial.
In summary, if x – 5 is a factor of f(x), then it must be true that:
- f(5) = 0 (5 is a root of the function).