Appearance
question:Given two polynomials in the variable ( x ) with integer coefficients. Their product is a polynomial in the variable ( x ) with even coefficients, not all of which are divisible by 4. Prove that in one polynomial all coefficients are even, while in the other polynomial at least one coefficient is odd.
answer:1. **Given**: Two polynomials in variable ( x ) with integer coefficients. The product of these two polynomials is a polynomial with even coefficients, not all of which are divisible by 4. 2. **To Prove**: One of the polynomials has all even coefficients, and the other has at least one odd coefficient. 3. **Assumption for Contradiction**: Suppose both polynomials have at least one odd coefficient. 4. **Reduction Modulo 2**: Replace each coefficient in both polynomials with its remainder when divided by 2. This essentially reduces our problem to working with polynomials over the field (mathbb{Z}_2), where coefficients are either 0 or 1. 5. **Representing Polynomials Modulo 2**: Let the two polynomials be ( P(x) ) and ( Q(x) ). Under modulo 2 reduction, these can be written as: [ P(x) = a_n x^n + a_{n-1} x^{n-1} + cdots + a_0 ] and [ Q(x) = b_m x^m + b_{m-1} x^{m-1} + cdots + b_0 ] where (a_i, b_i in {0, 1}) for all ( i ). 6. **Product**: The product of these polynomials modulo 2 is: [ P(x) cdot Q(x) = (a_n x^n + a_{n-1} x^{n-1} + cdots + a_0) cdot (b_m x^m + b_{m-1} x^{m-1} + cdots + b_0) ] This results in a polynomial: [ R(x) = c_{n+m} x^{n+m} + c_{n+m-1} x^{n+m-1} + cdots + c_0 ] where each ( c_k ) is the sum of products of the coefficients ( a_i cdot b_j ) such that (i + j = k), all calculated modulo 2. 7. **Analysis of Coefficients**: Since every coefficient in (P(x) cdot Q(x)) modulo 2 results from the sum of products of 0's and 1's, for any (c_k) to be even (0 modulo 2), an even number of odd products (a_i b_j = 1) must contribute to (c_k). 8. **Contradiction**: If both (P(x)) and (Q(x)) contain odd terms, there will be at least one coefficient (c_k = 1) in the product (R(x)), implying at least one term is odd modulo 2. This contradicts the given condition that all coefficients are even (0 modulo 2). 9. **Conclusion**: Therefore, our initial assumption that both polynomials can have odd coefficients must be false. This implies exactly one of the polynomials has all even coefficients, and the other has at least one odd coefficient. [ boxed{} ]
question:I wrote a long essay but I want to condense the writing to make it more concise. Here's the essay: The main cause of writing apprehension for many students is because they are fearful of criticism and are worried that their writing will not be considered good enough. This fear of criticism does not only stop students from writing but it can impact their motivation and self-esteem. It is especially difficult for students to be confident in themselves when they hear their peers discussing topics using complex vocabulary and concepts that they themselves do not understand. Furthermore, students may feel that they do not have the right or the authority to write about matters that have been discussed and critiqued by experts and scholars, thus promoting the idea that their opinions are not valid. It can be quite challenging for students to find their place in academic writing when it seems that every subject has been studied and discussed comprehensively. Additionally, students struggle with the idea that academic writing should be objective and absent of personal opinions and feelings, even though subjective influences are inevitable in any form of writing. Many students struggle to eliminate their voice from their work, while maintaining a unique and original perspective, leading to frustration and anxiety when writing. For many students, writing apprehension is a result of a combination of these factors which can consequently affect their self-perception, leading to lower academic achievements. With so many factors affecting students' perceptions of their writing capabilities, it can be quite challenging to overcome the anxiety and fear that many students face when writing academically. Can you help me condense this?
answer:I can help you make your essay more concise while maintaining its key points. Here's a condensed version: Many students experience writing apprehension due to a fear of criticism and the pressure to produce high-quality work. This fear not only hinders their writing but also impacts their motivation and self-esteem. The use of complex vocabulary and concepts by their peers can exacerbate these feelings of inadequacy. Students often feel they lack the authority to write about subjects already explored by experts, making it difficult for them to find their voice in academic writing. The expectation of objectivity in academic writing also poses a challenge, as students struggle to eliminate personal opinions while maintaining originality. These factors combine to affect students' self-perception, leading to lower academic achievements. To make your essay more concise, I removed some repetitive phrases and sentences, and condensed similar ideas into single sentences. I also focused on preserving the core arguments and emotions expressed in the original essay. The result is a more streamlined version that still conveys the main points and feelings of the original text.
question:How do I ensure objectivity and remove personal feelings from my essays while maintaining originality?
answer:Ensuring objectivity and removing personal feelings from essays can be challenging, but it's not about eliminating your unique perspective entirely. Instead, it's about striking a balance between presenting your arguments and supporting them with evidence, while avoiding biased language and emotional appeals. Here are some suggestions to help you achieve this: When presenting arguments, focus on using evidence-based statements instead of personal opinions. For example, instead of saying I believe that..., say According to research, it's evident that.... This way, you're still expressing your perspective, but grounding it in objective evidence. Use neutral language and avoid emotive tone. Phrases like clearly, obviously, or it's a fact can come across as biased. Instead, use more neutral phrases like studies suggest, research indicates, or data shows. To maintain originality, try to approach topics from a fresh angle or offer a nuanced perspective on an existing argument. Ask yourself questions like What's missing from the current discussion on this topic? or How can I challenge or build upon existing theories? Finally, acknowledge potential counterarguments and address them in your essay. This demonstrates that you've considered multiple perspectives and strengthens your argument. Remember, objectivity is not about suppressing your voice entirely, but about presenting your arguments in a clear, evidence-based, and balanced manner.
question:I also struggle with anticipating counterarguments in my essays. Can you offer any tips on how to anticipate and address these counterarguments effectively?
answer:Anticipating counterarguments is a crucial aspect of writing a strong essay. Here are some tips to help you anticipate and address counterarguments effectively: When researching your topic, actively seek out opposing views and opinions. Read articles, academic papers, and online discussions that present different perspectives on the issue. This will help you understand the potential counterarguments and prepare you to address them. Consider the potential flaws or weaknesses in your own argument. Ask yourself questions like What are the potential criticisms of my argument? or What are the limitations of my evidence? This will help you identify areas where counterarguments may arise. Put yourself in the shoes of someone who disagrees with your argument. Try to think like a skeptical reader and come up with potential objections to your points. This will help you anticipate counterarguments and prepare effective responses. When addressing counterarguments, acknowledge them explicitly in your essay. Use phrases like Some may argue that..., Critics might say..., or An opposing view is that.... Then, respond to the counterargument with evidence and reasoning that refutes or mitigates their concerns. Be sure to address counterarguments in a respectful and professional tone. Avoid being dismissive or condescending, as this can undermine your credibility. Instead, show that you've thoughtfully considered alternative perspectives and have a well-reasoned response. Finally, don't feel obligated to address every possible counterargument. Focus on the most significant and relevant ones, and prioritize addressing those that are most likely to be raised by your readers. By anticipating and addressing counterarguments, you can strengthen your argument, demonstrate your critical thinking skills, and show that you've thoroughly explored the topic.