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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: 📰 Youre Alone In A Digital Escape Gamewill You Outwit Traps Cracking Puzzles Before Midnight 1915149 📰 Can This Net Framework Repair Tool Fix Your Broken Code Overnight 9172756 📰 You Wont Believe What Happens When You Try Flex Pay 8201456 📰 Raji Green Bay 6054165 📰 Catbird Hotel 3586068 📰 Game Of Thrones Stark Robb Reveals Secrets No Fan Knewshocking Twist You Wont Believe 6801047 📰 5 The Forgotten Shortcut You Need To Take Screenshots Todayclick Here 5624309 📰 No One Saw Thisthe Silent Moment That Changed Everything Forever 9878443 📰 Ali Manno 7162546 📰 Hawaiian People 6664069 📰 Discover What Breaks The Mold Nude Africa Exposed Under The Sun 4992282 📰 But In The Absence Of Additional Constraints The Equation Is Identically Satisfied However For Olympiad Problems Often Such Questions Have Unique Answers So Perhaps Reconsider The Setup 6970440 📰 Akimbo Guns 5899059 📰 What Is An Ipo The Shocking Truth Behind How Companies Go Public 758204 📰 Futures Sp 500 6089312