We are given that [tex]\( f(c) = 0 \)[/tex]. Let's analyze each statement to determine which must be true.
Statement A: The point [tex]\((-c, 0)\)[/tex] lies on the graph of [tex]\(f(x)\)[/tex].
This statement is incorrect. The notation [tex]\( f(c) = 0 \)[/tex] implies that when [tex]\( x = c \)[/tex], [tex]\( f(x) = 0 \)[/tex]. Therefore, the correct point on the graph of [tex]\( f(x) \)[/tex] would be [tex]\((c, 0)\)[/tex], not [tex]\((-c, 0)\)[/tex].
Statement B: [tex]\( x + c \)[/tex] divides evenly into [tex]\( f(x) \)[/tex].
This statement is incorrect. The fact that [tex]\( f(c) = 0 \)[/tex] implies that [tex]\( c \)[/tex] is a root of the polynomial [tex]\( f(x) \)[/tex]. Therefore, [tex]\( f(x) \)[/tex] can be factored to include [tex]\( (x - c) \)[/tex] as a factor, not [tex]\( (x + c) \)[/tex].
Statement C: [tex]\( x - c \)[/tex] is a factor of [tex]\( f(x) \)[/tex].
This statement is correct. Given that [tex]\( f(c) = 0 \)[/tex], it means that [tex]\( c \)[/tex] is a root of the polynomial [tex]\( f(x) \)[/tex], and therefore, [tex]\( (x - c) \)[/tex] must be a factor of [tex]\( f(x) \)[/tex]. This is consistent with the factor theorem, which states that if a polynomial [tex]\( f(x) \)[/tex] has a root at [tex]\( x = c \)[/tex], then [tex]\( (x - c) \)[/tex] is a factor of the polynomial.
Statement D: The point [tex]\( (0, c) \)[/tex] lies on the graph of [tex]\( f(x) \)[/tex].
This statement is incorrect. [tex]\( f(c) = 0 \)[/tex] means that the function [tex]\( f(x) \)[/tex] evaluates to 0 when [tex]\( x = c \)[/tex], indicating the presence of the point [tex]\( (c, 0) \)[/tex]. The point [tex]\( (0, c) \)[/tex] would imply that the function evaluates to [tex]\( c \)[/tex] when [tex]\( x = 0 \)[/tex], which is not what the given information states.
Thus, the correct statement is:
C. [tex]\( x - c \)[/tex] is a factor of [tex]\( f(x) \)[/tex].
