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📰 eq 0 $. Contradiction? Wait, from $ k(2) = 0 $, check $ x = 1, y = -1 $: $ k(0) = k(1) + k(-1) - 2k(-1) = 1 + k(-1) - 2k(-1) = 1 - k(-1) $. Also $ k(0) = k(0 + 0) = 2k(0) - 2k(0) = 0 $? No: $ k(0) = k(0 + 0) = 2k(0) - 2k(0) = 0 $. So $ k(0) = 0 $. Then $ 0 = 1 - k(-1) $ → $ k(-1) = 1 $. Then $ x = -1, y = -1 $: $ k(-2) = 2k(-1) - 2k(1) = 2(1) - 2(1) = 0 $. $ x = 1, y = -1 $: $ k(0) = k(1) + k(-1) - 2k(-1) = 1 + 1 - 2(1) = 0 $, consistent. Now $ x = 2, y = -1 $: $ k(1) = k(2) + k(-1) - 2k(-2) = 0 + 1 - 0 = 1 $, matches. No contradiction. Thus $ k(2) = 0 $. Final answer: $ oxed{0} $. 📰 Question: Find the remainder when $ x^5 - 3x^3 + 2x - 1 $ is divided by $ x^2 - 2x + 1 $. 📰 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? 📰 Total Depth 12 05 6 Meters Days Required 6 3 2 Days 📰 Total Distance 150 200 350 Miles 📰 Total Distance Is 120 180 300 Miles 📰 Total Distance Is 150 200 350 Km 📰 Total Distinct Species Wetland Only Forest Only Both 📰 Total Energy 500 5 2500 Kcal 📰 Total Errors 2500 2300 2500 2300200200 📰 Total For 6 Turns 6 Times 1181 Approx 7086 M 📰 Total From Monday To Saturday 225 825 22582530753075 Pages 📰 Total Grant 1200000 📰 Total Growing Species 34 23 18 75 📰 Total Hours At Current Combined Rate 15 Hoursday 60 Days 900 Hours But This Is More Than Needed We Need To Compute Actual Hours With Proportional Work 📰 Total Hours Completed 30 20 15 65 Hours 📰 Total In 8 Months 10 Times 8 80 Badges 📰 Total Moles 009 010 019 Textmol