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8. All polynomials in P, such that p(0) = 0.

8. All polynomials in P, such that p(0) = 0.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Exercise Instructions: **In Exercises 5-8**, determine if the given set is a subspace of \( \mathbb{P}_n \) for an appropriate value of \( n \). **Justify your answers.** --- **Explanation:** In this set of exercises, you are asked to evaluate whether certain sets of polynomial functions form subspaces of the vector space \( \mathbb{P}_n \), where \( \mathbb{P}_n \) denotes the vector space of all polynomials of degree at most \( n \). Your task is to: 1. **Identify the Set**: Examine the given set of polynomials. 2. **Determine the Subspace**: Verify if this set meets the criteria to be a subspace of \( \mathbb{P}_n \). 3. **Justify Your Answer**: Provide a clear and mathematical justification for your conclusion. A set is a subspace of \( \mathbb{P}_n \) if: - It includes the zero polynomial. - It is closed under polynomial addition. - It is closed under scalar multiplication. Use these criteria to guide your analysis and justification.
Transcribed Image Text:### Exercise Instructions: **In Exercises 5-8**, determine if the given set is a subspace of \( \mathbb{P}_n \) for an appropriate value of \( n \). **Justify your answers.** --- **Explanation:** In this set of exercises, you are asked to evaluate whether certain sets of polynomial functions form subspaces of the vector space \( \mathbb{P}_n \), where \( \mathbb{P}_n \) denotes the vector space of all polynomials of degree at most \( n \). Your task is to: 1. **Identify the Set**: Examine the given set of polynomials. 2. **Determine the Subspace**: Verify if this set meets the criteria to be a subspace of \( \mathbb{P}_n \). 3. **Justify Your Answer**: Provide a clear and mathematical justification for your conclusion. A set is a subspace of \( \mathbb{P}_n \) if: - It includes the zero polynomial. - It is closed under polynomial addition. - It is closed under scalar multiplication. Use these criteria to guide your analysis and justification.
**8. All polynomials in \( \mathbb{P}_n \) such that \( p(0) = 0 \).** In this question, we are considering a specific subset of polynomials within the polynomial space \( \mathbb{P}_n \). The condition \( p(0) = 0 \) implies that these polynomials must have zero as one of their roots. This means any polynomial \( p(x) \) in this subset can be expressed as \( x \cdot q(x) \), where \( q(x) \) is a polynomial of degree \( n-1 \). For example, consider the space \( \mathbb{P}_3 \) which includes all polynomials of degree at most 3. Polynomials in \( \mathbb{P}_3 \) satisfying \( p(0) = 0 \) could have forms such as: - \( p(x) = x(x^2 - 3x + 2) \), which simplifies to \( p(x) = x^3 - 3x^2 + 2x \). - \( p(x) = x(x + 1)^2 \), which simplifies to \( p(x) = x^3 + 2x^2 + x \). In general, any polynomial \( p(x) = c_n x^n + c_{n-1} x^{n-1} + \cdots + c_1 x \) in \( \mathbb{P}_n \) that meets \( p(0) = 0 \) can be written with \( c_0 = 0 \). No graphs or diagrams are included with this text.
Transcribed Image Text:**8. All polynomials in \( \mathbb{P}_n \) such that \( p(0) = 0 \).** In this question, we are considering a specific subset of polynomials within the polynomial space \( \mathbb{P}_n \). The condition \( p(0) = 0 \) implies that these polynomials must have zero as one of their roots. This means any polynomial \( p(x) \) in this subset can be expressed as \( x \cdot q(x) \), where \( q(x) \) is a polynomial of degree \( n-1 \). For example, consider the space \( \mathbb{P}_3 \) which includes all polynomials of degree at most 3. Polynomials in \( \mathbb{P}_3 \) satisfying \( p(0) = 0 \) could have forms such as: - \( p(x) = x(x^2 - 3x + 2) \), which simplifies to \( p(x) = x^3 - 3x^2 + 2x \). - \( p(x) = x(x + 1)^2 \), which simplifies to \( p(x) = x^3 + 2x^2 + x \). In general, any polynomial \( p(x) = c_n x^n + c_{n-1} x^{n-1} + \cdots + c_1 x \) in \( \mathbb{P}_n \) that meets \( p(0) = 0 \) can be written with \( c_0 = 0 \). No graphs or diagrams are included with this text.
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