To determine if the statement "When the polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( (x - a) \)[/tex], the remainder equals [tex]\( P(a) \)[/tex]" is true or false, let's explore it step-by-step.
### Step-by-Step Solution
1. Understanding Polynomial Division:
- Let's consider a polynomial [tex]\( P(x) \)[/tex].
- When [tex]\( P(x) \)[/tex] is divided by a linear divisor of the form [tex]\( (x - a) \)[/tex], it can be expressed as:
[tex]\[
P(x) = (x - a)Q(x) + R
\][/tex]
where [tex]\( Q(x) \)[/tex] is the quotient and [tex]\( R \)[/tex] is the remainder.
2. Considering the Remainder:
- Since [tex]\( x - a \)[/tex] is a linear divisor (a polynomial of degree 1), the degree of the remainder [tex]\( R \)[/tex] must be less than the degree of the divisor. Therefore, [tex]\( R \)[/tex] is a constant.
3. Applying the Remainder Theorem:
- The Remainder Theorem states that for any polynomial [tex]\( P(x) \)[/tex], when divided by [tex]\( (x-a) \)[/tex], the remainder of this division is [tex]\( P(a) \)[/tex].
- This means substituting [tex]\( x = a \)[/tex] in [tex]\( P(x) \)[/tex] will yield the remainder:
[tex]\[
R = P(a)
\][/tex]
4. Verifying the Statement:
- Given the properties of the polynomial division and the Remainder Theorem, the statement "When the polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( (x-a) \)[/tex], the remainder equals [tex]\( P(a) \)[/tex]" is evaluated.
### Conclusion
The statement is indeed True. When the polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( (x-a) \)[/tex], the remainder is exactly [tex]\( P(a) \)[/tex].
So, the correct answer is A. True.
