Let's delve into the problem step-by-step to understand whether the statement is true or false:
1. Understanding Polynomials and Remainders:
When a polynomial [tex]\( f(x) \)[/tex] is divided by another polynomial, there may be a remainder. For instance, if you divide [tex]\( f(x) \)[/tex] by a linear polynomial [tex]\( (x-a) \)[/tex], the process involves polynomial long division.
2. The Factor Theorem:
The Factor Theorem is a key concept here. It states:
- "A polynomial [tex]\( f(x) \)[/tex] has a factor [tex]\( (x-a) \)[/tex] if and only if [tex]\( f(a) = 0 \)[/tex]."
This means that if substituting [tex]\( x = a \)[/tex] into the polynomial [tex]\( f(x) \)[/tex] results in zero, then [tex]\( (x-a) \)[/tex] is a factor of [tex]\( f(x) \)[/tex].
3. Connecting the Remainder to the Factor:
When you divide a polynomial [tex]\( f(x) \)[/tex] by [tex]\( (x-a) \)[/tex] and the remainder is zero, it shows that the polynomial [tex]\( f(x) \)[/tex] can be expressed as [tex]\( f(x) = (x-a)q(x) \)[/tex], where [tex]\( q(x) \)[/tex] is the quotient polynomial. This also means that [tex]\( f(a) = 0 \)[/tex], implying [tex]\( a \)[/tex] is a root of [tex]\( f(x) \)[/tex].
4. Conclusion:
Given the Factor Theorem, if the remainder when [tex]\( f(x) \)[/tex] is divided by [tex]\( (x-a) \)[/tex] is zero, [tex]\( (x-a) \)[/tex] must be a factor of [tex]\( f(x) \)[/tex]. Conversely, if [tex]\( (x-a) \)[/tex] is a factor of [tex]\( f(x) \)[/tex], then dividing [tex]\( f(x) \)[/tex] by [tex]\( (x-a) \)[/tex] will yield a remainder of zero.
Therefore, 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 proven true by the Factor Theorem.
The answer is:
A. True
