Welcome to Westonci.ca, your go-to destination for finding answers to all your questions. Join our expert community today! Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
Let's analyze the statement and determine its correctness step by step.
### Step-by-Step Analysis
1. Remainder Theorem: The Remainder Theorem states that if a polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( (x - c) \)[/tex], then the remainder of this division is [tex]\( P(c) \)[/tex]. This theorem provides a direct way to find the remainder without performing the full polynomial division.
2. Given Expression: The statement in the problem is that the polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( (x + a) \)[/tex]. This expression can be rewritten by recognizing that [tex]\( (x + a) \)[/tex] is the same as [tex]\( (x - (-a)) \)[/tex].
3. Applying the Remainder Theorem:
- According to the Remainder Theorem, for the polynomial [tex]\( P(x) \)[/tex] divided by [tex]\( (x + a) \)[/tex], we would evaluate the polynomial at [tex]\( -a \)[/tex], because [tex]\( (x + a) \)[/tex] can be rewritten as [tex]\( (x - (-a)) \)[/tex].
- Therefore, the remainder when [tex]\( P(x) \)[/tex] is divided by [tex]\( (x + a) \)[/tex] is [tex]\( P(-a) \)[/tex].
4. Given Statement: The given statement claims that the remainder equals [tex]\( P(a) \)[/tex]. According to our analysis, the correct answer should be [tex]\( P(-a) \)[/tex], not [tex]\( P(a) \)[/tex].
### Conclusion
Based on the steps above, the correct remainder when the polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( (x + a) \)[/tex] should be [tex]\( P(-a) \)[/tex]. Hence, the given statement "the remainder equals [tex]\( P(a) \)[/tex]" is incorrect.
Therefore, the correct answer is:
B. False
### Step-by-Step Analysis
1. Remainder Theorem: The Remainder Theorem states that if a polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( (x - c) \)[/tex], then the remainder of this division is [tex]\( P(c) \)[/tex]. This theorem provides a direct way to find the remainder without performing the full polynomial division.
2. Given Expression: The statement in the problem is that the polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( (x + a) \)[/tex]. This expression can be rewritten by recognizing that [tex]\( (x + a) \)[/tex] is the same as [tex]\( (x - (-a)) \)[/tex].
3. Applying the Remainder Theorem:
- According to the Remainder Theorem, for the polynomial [tex]\( P(x) \)[/tex] divided by [tex]\( (x + a) \)[/tex], we would evaluate the polynomial at [tex]\( -a \)[/tex], because [tex]\( (x + a) \)[/tex] can be rewritten as [tex]\( (x - (-a)) \)[/tex].
- Therefore, the remainder when [tex]\( P(x) \)[/tex] is divided by [tex]\( (x + a) \)[/tex] is [tex]\( P(-a) \)[/tex].
4. Given Statement: The given statement claims that the remainder equals [tex]\( P(a) \)[/tex]. According to our analysis, the correct answer should be [tex]\( P(-a) \)[/tex], not [tex]\( P(a) \)[/tex].
### Conclusion
Based on the steps above, the correct remainder when the polynomial [tex]\( P(x) \)[/tex] is divided by [tex]\( (x + a) \)[/tex] should be [tex]\( P(-a) \)[/tex]. Hence, the given statement "the remainder equals [tex]\( P(a) \)[/tex]" is incorrect.
Therefore, the correct answer is:
B. False
Thank you for visiting our platform. We hope you found the answers you were looking for. Come back anytime you need more information. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Westonci.ca is committed to providing accurate answers. Come back soon for more trustworthy information.