Let's analyze the given statement carefully to determine if it is true or false:
"A number [tex]\( a \)[/tex] is a root of [tex]\( P(x) \)[/tex] if and only if the remainder, when dividing the polynomial by [tex]\( x-a \)[/tex], equals zero."
We need to understand what it means for [tex]\( a \)[/tex] to be a root of a polynomial [tex]\( P(x) \)[/tex] and what it means for the remainder when dividing [tex]\( P(x) \)[/tex] by [tex]\( x-a \)[/tex].
1. A number [tex]\( a \)[/tex] is a root of the polynomial [tex]\( P(x) \)[/tex] if [tex]\( P(a) = 0 \)[/tex]. This means that when we substitute [tex]\( a \)[/tex] into the polynomial [tex]\( P(x) \)[/tex], the result is zero.
2. When dividing the polynomial [tex]\( P(x) \)[/tex] by the binomial [tex]\( x-a \)[/tex], according to polynomial division (remainder theorem), the remainder of this division is exactly [tex]\( P(a) \)[/tex]. If [tex]\( a \)[/tex] is a root of [tex]\( P(x) \)[/tex], then [tex]\( P(a) = 0 \)[/tex].
Therefore, if [tex]\( a \)[/tex] is a root of [tex]\( P(x) \)[/tex], the remainder when dividing [tex]\( P(x) \)[/tex] by [tex]\( x-a \)[/tex] will be zero. Conversely, if the remainder is zero when [tex]\( P(x) \)[/tex] is divided by [tex]\( x-a \)[/tex], it indicates that [tex]\( P(a) = 0 \)[/tex], meaning [tex]\( a \)[/tex] is indeed a root of [tex]\( P(x) \)[/tex].
Thus, the statement:
"A number [tex]\( a \)[/tex] is a root of [tex]\( P(x) \)[/tex] if and only if the remainder, when dividing the polynomial by [tex]\( x-a \)[/tex], equals zero." is indeed true.
So the correct answer is:
A. True
