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📰 Solution: Note $ x^2 - 2x + 1 = (x - 1)^2 $. Use polynomial division or remainder theorem for repeated roots. Let $ f(x) = x^5 - 3x^3 + 2x - 1 $. The remainder $ R(x) $ has degree < 2, so $ R(x) = ax + b $. Since $ (x - 1)^2 $ divides $ f(x) - R(x) $, we have $ f(1) = R(1) $ and $ f'(1) = R'(1) $. Compute $ f(1) = 1 - 3 + 2 - 1 = -1 $. $ f'(x) = 5x^4 - 9x^2 + 2 $, so $ f'(1) = 5 - 9 + 2 = -2 $. $ R(x) = ax + b $, so $ R(1) = a + b = -1 $, $ R'(x) = a $, so $ a = -2 $. Then $ -2 + b = -1 $ → $ b = 1 $. Thus, remainder is $ -2x + 1 $. Final answer: $ oxed{-2x + 1} $.Question: A plant biologist is studying a genetic trait that appears in every 12th plant in a rows of crops planted in a 120-plant grid. If the trait is expressed only when the plant’s position number is relatively prime to 12, how many plants in the first 120 positions exhibit the trait? 📰 Solution: We are to count how many integers from 1 to 120 are relatively prime to 12. This is given by Euler’s totient function applied to the set of positions, but since the condition is relative primality with 12, we compute $ \phi(12) $, the number of integers from 1 to 12 that are coprime to 12, and then multiply by the number of full cycles in 120. 📰 First, factor 12: 📰 Shellpoint Breakthrough The Most Hyped Feature You Cant Afford To Miss 2850557 📰 How Many Pounds Are 4 Kilos 1912805 📰 Canopy Growth Corp Share Price 4848780 📰 Witcher Season 5 6353473 📰 Sorry Youre Missing Outheres How To Instantly Share Your Calendar In Outlook 735240 📰 Boom In Design Precision How Cad Bane Outshines All Competitors In 2024 6078828 📰 Race In Spanish Language 2830609 📰 Your Imessage Will Work Like Magicyoure About To Discover Why 3140876 📰 Found The Lord In The Darknesshis Voice Was Never Silent 5520736 📰 Wfm Stock Price 9222150 📰 United Health Stock 8978945 📰 The Shocking Truth Behind Upstairs Circus You Wont Believe What Happened Next 2514317 📰 Airdroid Cast The Secret Tool Making Your Phone Run Android Cast Like A Pro 1523322 📰 Kevin Bacon New Show 4581337 📰 Add Bang To Your Brand With These Trendy Coffee Clipart Imagesfree Eye Catching 7110619