Let's analyze the statement and determine its correctness step by step.
### Step-by-Step Analysis
1. Remainder Theorem: The Remainder Theorem states that if a polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( (x - c) \)[/tex], then the remainder of this division is [tex]\( P(c) \)[/tex]. This theorem provides a direct way to find the remainder without performing the full polynomial division.
2. Given Expression: The statement in the problem is that the polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( (x + a) \)[/tex]. This expression can be rewritten by recognizing that [tex]\( (x + a) \)[/tex] is the same as [tex]\( (x - (-a)) \)[/tex].
3. Applying the Remainder Theorem:
- According to the Remainder Theorem, for the polynomial [tex]\( P(x) \)[/tex] divided by [tex]\( (x + a) \)[/tex], we would evaluate the polynomial at [tex]\( -a \)[/tex], because [tex]\( (x + a) \)[/tex] can be rewritten as [tex]\( (x - (-a)) \)[/tex].
- Therefore, the remainder when [tex]\( P(x) \)[/tex] is divided by [tex]\( (x + a) \)[/tex] is [tex]\( P(-a) \)[/tex].
4. Given Statement: The given statement claims that the remainder equals [tex]\( P(a) \)[/tex]. According to our analysis, the correct answer should be [tex]\( P(-a) \)[/tex], not [tex]\( P(a) \)[/tex].
### Conclusion
Based on the steps above, the correct remainder when the polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( (x + a) \)[/tex] should be [tex]\( P(-a) \)[/tex]. Hence, the given statement "the remainder equals [tex]\( P(a) \)[/tex]" is incorrect.
Therefore, the correct answer is:
B. False
