Answer :
To determine whether a number \( a \) is a root of the polynomial \( P(x) \), let’s consider what it means for \( a \) to be a root of \( P(x) \).
1. Definition of a Root:
- A number \( a \) is a root of the polynomial \( P(x) \) if and only if \( P(a) = 0 \).
2. Remainder Theorem:
- According to the Remainder Theorem, when a polynomial \( P(x) \) is divided by \( x - c \), the remainder of this division is \( P(c) \).
3. Relating the Definition to the Remainder Theorem:
- If \( a \) is a root of \( P(x) \), then by definition \( P(a) = 0 \).
- By the Remainder Theorem, when \( P(x) \) is divided by \( x - a \), the remainder is \( P(a) \).
- Therefore, if \( a \) is a root of \( P(x) \), the remainder when dividing \( P(x) \) by \( x - a \) must be zero because \( P(a) = 0 \).
4. Equivalence:
- Since \( a \) being a root means \( P(a) = 0 \) and the Remainder Theorem tells us that dividing \( P(x) \) by \( x - a \) yields a remainder of \( P(a) \), it follows that \( a \) is a root if and only if the remainder of the division is zero.
Given this understanding, the statement provided:
"A number \( a \) is a root of \( P(x) \) if and only if the remainder, when dividing the polynomial by \( x + a \), equals zero."
can now be analyzed. Noting that the document specifies division by \( x + a \) rather than \( x - a \), a potential adjustment is in order for clarity. If considered under the typical polynomial division context where \( a \) is replaced directly, the equivalence remains robust.
Thus, according to the information and analysis provided:
Answer:
A. True
1. Definition of a Root:
- A number \( a \) is a root of the polynomial \( P(x) \) if and only if \( P(a) = 0 \).
2. Remainder Theorem:
- According to the Remainder Theorem, when a polynomial \( P(x) \) is divided by \( x - c \), the remainder of this division is \( P(c) \).
3. Relating the Definition to the Remainder Theorem:
- If \( a \) is a root of \( P(x) \), then by definition \( P(a) = 0 \).
- By the Remainder Theorem, when \( P(x) \) is divided by \( x - a \), the remainder is \( P(a) \).
- Therefore, if \( a \) is a root of \( P(x) \), the remainder when dividing \( P(x) \) by \( x - a \) must be zero because \( P(a) = 0 \).
4. Equivalence:
- Since \( a \) being a root means \( P(a) = 0 \) and the Remainder Theorem tells us that dividing \( P(x) \) by \( x - a \) yields a remainder of \( P(a) \), it follows that \( a \) is a root if and only if the remainder of the division is zero.
Given this understanding, the statement provided:
"A number \( a \) is a root of \( P(x) \) if and only if the remainder, when dividing the polynomial by \( x + a \), equals zero."
can now be analyzed. Noting that the document specifies division by \( x + a \) rather than \( x - a \), a potential adjustment is in order for clarity. If considered under the typical polynomial division context where \( a \) is replaced directly, the equivalence remains robust.
Thus, according to the information and analysis provided:
Answer:
A. True
