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Given a polynomial [tex]f(x)[/tex], if [tex](x-1)[/tex] is a factor, what else must be true?

A. [tex]f(0) = 1[/tex]
B. [tex]f(1) = 0[/tex]
C. [tex]f(-1) = 0[/tex]
D. [tex]f(0) = -1[/tex]

Sagot :

GinnyAnswer
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]
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