To determine whether the statement "If a polynomial is divided by [tex]\((x - a)\)[/tex] and the remainder equals zero, then [tex]\((x - a)\)[/tex] is a factor of the polynomial" is true or false, we need to delve into the Factor Theorem.
The Factor Theorem is an important concept in algebra that connects the roots of a polynomial to its factors. The theorem states that:
1. A polynomial [tex]\(P(x)\)[/tex] has a factor [tex]\((x - a)\)[/tex] if and only if [tex]\(P(a) = 0\)[/tex].
2. If [tex]\(P(a) = 0\)[/tex], then the polynomial [tex]\(P(x)\)[/tex] can be expressed as [tex]\(P(x) = (x - a)Q(x)\)[/tex], where [tex]\(Q(x)\)[/tex] is another polynomial.
Given this theorem, let's analyze the given statement:
- When a polynomial [tex]\(P(x)\)[/tex] is divided by [tex]\((x - a)\)[/tex] and the remainder is zero, this means that substituting [tex]\(x = a\)[/tex] into the polynomial [tex]\(P(x)\)[/tex] results in [tex]\(P(a) = 0\)[/tex].
- According to the Factor Theorem, if [tex]\(P(a) = 0\)[/tex], then [tex]\((x - a)\)[/tex] is a factor of [tex]\(P(x)\)[/tex].
Therefore, if the remainder is zero when [tex]\(P(x)\)[/tex] is divided by [tex]\((x - a)\)[/tex], [tex]\((x - a)\)[/tex] must be a factor of the polynomial [tex]\(P(x)\)[/tex].
Given this reasoning, the correct answer to the statement is:
A. True
