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question:A polynomial q(x) leaves a remainder of 5 when divided by x - 3, a remainder of -2 when divided by x - 4, and a remainder of 6 when divided by x + 2. Let s(x) be the remainder when q(x) is divided by (x - 3)(x - 4)(x + 2). Find s(5).
answer:By the Remainder Theorem, q(3) = 5, q(4) = -2, and q(-2) = 6. When q(x) is divided by (x - 3)(x - 4)(x + 2), the remainder is of the form ax^2 + bx + c. Thus, [q(x) = (x - 3)(x - 4)(x + 2) r(x) + ax^2 + bx + c] for some polynomial r(x). Setting x = 3, x = 4, and x = -2, we get begin{align*} 9a + 3b + c &= q(3) = 5, 16a + 4b + c &= q(4) = -2, 4a - 2b + c &= q(-2) = 6. end{align*} Subtracting these equations in pairs, we find: begin{align*} 7a + b &= -7, 12a + 6b &= -8. end{align*} Dividing the second equation by 6 gives 2a + b = -frac{4}{3}. Solving these two equations simultaneously, we find: [ 7a + b = -7 2a + b = -frac{4}{3} ] Subtracting these gives 5a = -frac{17}{3}, so a = -frac{17}{15}. Substituting back, b = -frac{4}{3} - 2(-frac{17}{15}) = -frac{4}{3} + frac{34}{15} = -frac{26}{45}. Using 9a + 3b + c = 5, c = 5 - 9(-frac{17}{15}) - 3(-frac{26}{45}) = 5 + frac{153}{15} - frac{78}{45} = 5 + frac{51}{5} - frac{26}{15} = frac{63}{15} = 4.2. Thus, s(x) = -frac{17}{15}x^2 - frac{26}{45}x + 4.2. Evaluating s(5): [s(5) = -frac{17}{15} cdot 25 - frac{26}{45} cdot 5 + 4.2 = -frac{425}{15} - frac{130}{45} + 4.2 = -28.33 - 2.89 + 4.2 = -27.02] Thus, s(5) approx boxed{-27}.
question:Asian Americans in New York City, according to the 2010 Census, number more than one million, greater than the combined totals of San Francisco and Los Angeles. New York contains the highest total Asian population of any U.S. city proper. The New York City borough of Queens is home to the state's largest Asian American population and the largest Andean (Colombian, Ecuadorian, Peruvian, and Bolivian) populations in the United States, and is also the most ethnically diverse urban area in the world. The Chinese population constitutes the fastest-growing nationality in New York State; multiple satellites of the original Manhattan Chinatown (紐約華埠), in Brooklyn (布鲁克林華埠), and around Flushing, Queens (法拉盛華埠), are thriving as traditionally urban enclaves, while also expanding rapidly eastward into suburban Nassau County (拿騷縣) on Long Island (長島), as the New York metropolitan region and New York State have become the top destinations for new Chinese immigrants, respectively, and large-scale Chinese immigration continues into New York City and surrounding areas. In 2012, 6.3% of New York City was of Chinese ethnicity, with nearly three-fourths living in either Queens or Brooklyn, geographically on Long Island. A community numbering 20,000 Korean-Chinese (Chaoxianzu (Chinese: 朝鲜族) or Joseonjok (Hangul: 조선족)) is centered in Flushing, Queens, while New York City is also home to the largest Tibetan population outside China, India, and Nepal, also centered in Queens. Koreans made up 1.2% of the city's population, and Japanese 0.3%. Filipinos were the largest Southeast Asian ethnic group at 0.8%, followed by Vietnamese, who made up 0.2% of New York City's population in 2010. Indians are the largest South Asian group, comprising 2.4% of the city's population, with Bangladeshis and Pakistanis at 0.7% and 0.5%, respectively. Queens is the preferred borough of settlement for Asian Indians, Koreans, and Filipinos, as well as Malaysians and other Southeast Asians; while Brooklyn is receiving large numbers of both West Indian as well as Asian Indian immigrants. Answer this question, if possible (if impossible, reply unanswerable): What New York borough contains the highest population of Asian-Americans?
answer:The borough in New York that has the highest population of Asian-Americans is Queens. Just think of it like Queens is the most popular neighborhood where many Asian-Americans like to live!
question:Can you guide me through the process of changing my profile picture on LinkedIn?
answer:Of course, I'd be happy to help you change your profile picture on LinkedIn. Here are the steps: 1. Log in to your LinkedIn account. 2. Once you're logged in, click on the 'Me' icon located at the top right of your LinkedIn homepage. 3. A drop-down menu will appear. Click on the 'View Profile' option. 4. On your profile page, hover over your profile photo. You'll see a camera icon appear on the photo. 5. Click on this camera icon. 6. A pop-up window will appear, click on the 'Change photo' button. 7. You'll then be prompted to select a photo from your device. Choose the photo you want to use as your profile picture. 8. After uploading, you can adjust the position and size of the image using LinkedIn's editing tool. 9. Once you're satisfied with how your picture looks, click on 'Save photo'. 10. Your LinkedIn profile picture should now be updated. Remember, LinkedIn suggests using a photo that represents the professional you, and recommends a headshot with a clear view of your face. The recommended size for a LinkedIn profile picture is 400 x 400 pixels.
question:Hello I'm going to submit some paragraphs to you and I would like to edit them to make them more concise and impactful can you give me some feedback?
answer:I'd be happy to help you refine your writing. Please go ahead and share the paragraphs you'd like me to review, and I'll provide you with constructive feedback on how to make them more concise and impactful. I'll suggest edits, rephrase sentences, and offer ideas to improve clarity and overall flow. Let's get started!