Sure! Let's break down the problem step-by-step!
Given a polynomial [tex]\( f(x) \)[/tex] with [tex]\((x + 3)\)[/tex] as a factor, we need to determine which of the following must be true:
1. [tex]\( f(-3) = 0 \)[/tex]
2. [tex]\( f(0) = -3 \)[/tex]
3. [tex]\( f(3) = 0 \)[/tex]
4. [tex]\( f(0) = 3 \)[/tex]
### Analyzing the Factor:
When a polynomial [tex]\( f(x) \)[/tex] has a factor [tex]\((x + 3)\)[/tex], this means that [tex]\( f(x) \)[/tex] can be expressed as:
[tex]\[ f(x) = (x + 3) \cdot q(x) \][/tex]
where [tex]\( q(x) \)[/tex] is another polynomial.
### Factor Theorem:
The factor theorem states that if [tex]\((x + 3)\)[/tex] is a factor of [tex]\( f(x) \)[/tex], then substituting [tex]\(-3\)[/tex] for [tex]\( x \)[/tex] in [tex]\( f(x) \)[/tex] will yield zero. This is because for any value [tex]\( a \)[/tex], if [tex]\((x - a)\)[/tex] is a factor of [tex]\( f(x) \)[/tex], then [tex]\( f(a) = 0 \)[/tex].
### Substitution:
In this specific problem, [tex]\((x + 3)\)[/tex] implies [tex]\( a = -3 \)[/tex]. Accordingly, [tex]\((x - (-3)) = (x + 3)\)[/tex]. Therefore:
[tex]\[ f(-3) = 0 \][/tex]
### Conclusion:
Thus, the correct statement must be:
1. [tex]\( f(-3) = 0 \)[/tex].
So, if [tex]\((x + 3)\)[/tex] is a factor of the polynomial [tex]\( f(x) \)[/tex], it must be true that [tex]\( f(-3) = 0 \)[/tex]. This satisfies the factor theorem and confirms our understanding of polynomial roots and factors.
Hence, the correct answer is:
[tex]\( f(-3) = 0 \)[/tex].
