To solve the given problem, we need to recall an important property of polynomials and factors.
If [tex]\((x - 1)\)[/tex] is a factor of a polynomial [tex]\(f(x)\)[/tex], then by the Factor Theorem, it means that the polynomial [tex]\(f(x)\)[/tex] will have a root at [tex]\(x = 1\)[/tex]. In other words, the polynomial will evaluate to zero when [tex]\(x = 1\)[/tex].
Step-by-step:
1. Understanding the Factor Theorem:
The Factor Theorem states that a polynomial [tex]\(f(x)\)[/tex] has a factor [tex]\((x - c)\)[/tex] if and only if [tex]\(f(c) = 0\)[/tex].
2. Applying the Factor Theorem:
- Given: [tex]\((x - 1)\)[/tex] is a factor of [tex]\(f(x)\)[/tex].
- By the Factor Theorem: [tex]\(f(1) = 0\)[/tex].
3. Evaluating the Property:
- Since [tex]\( (x - 1) \)[/tex] is a factor, it must be true that substituting [tex]\(x = 1\)[/tex] into [tex]\(f(x)\)[/tex] results in zero.
Thus, the statement that must be true if [tex]\((x - 1)\)[/tex] is a factor of the polynomial [tex]\(f(x)\)[/tex] is:
[tex]\[ f(1) = 0 \][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{f(1) = 0} \][/tex]
