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 $ → $

["Solution: How Many Plants Exhibit the Genetic Trait in a 120-Plant Grid?", "In a carefully arranged 120-plant grid, a rare genetic trait appears in every 12th plant—specifically at positions corresponding to multiples of 12. However, the trait manifests only in those plant positions that are relatively prime to 12. To determine how many plants in the first 120 positions truly exhibit the trait, we must count the integers from 1 to 120 that are relatively prime to 12.", "### Understanding Relative Primality", "Two numbers are relatively prime if their greatest common divisor (gcd) is 1. Since 12 factors as $ 12 = 2^2 \cdot 3 $, a number is relatively prime to 12 if it is not divisible by 2 or 3.", "We use the Inclusion-Exclusion Principle to count how many integers from 1 to 120 are divisible by 2 or 3, then subtract from 120 to find those coprime to 12.", "Let:\n- $ A = $ multiples of 2 in [1, 120]\n- $ B = $ multiples of 3 in [1, 120]\n- $ A \cap B = $ multiples of $ \ ext{lcm}(2,3) = 6 $", "Then:\n- $ |A| = \left\lfloor \frac{120}{2} \right\rfloor = 60 $\n- $ |B| = \left\lfloor \frac{120}{3} \right\rfloor = 40 $\n- $ |A \cap B| = \left\lfloor \frac{120}{6} \right\rfloor = 20 $", "By inclusion-exclusion:\n$$\n|A \cup B| = |A| + |B| - |A \cap B| = 60 + 40 - 20 = 80\n$$", "So, 80 numbers are divisible by 2 or 3 → they are not relatively prime to 12.", "Therefore, the count of numbers relatively prime to 12 is:\n$$\n120 - 80 = 40\n$$", "### Connection to Polynomial Division (Conceptual Insight)", "Interestingly, this problem of counting coprime integers echoes deeper number-theoretic principles seen in modular arithmetic and polynomial identities—such as using the remainder when dividing by $ x - 1 $, inspired by the Remainder Theorem. For instance, the degree-restricted remainder $ R(x) = ax + b $ for step functions mirrors how inverse roots reveal periodic structure—just as $ (x - 1)^2 $ divides a polynomial at $ x = 1 $, only non-roots (like values coprime to 12) sustain non-vanishing behavior across intervals.", "### Final Answer", "The number of plants in the first 120 positions that exhibit the trait—those whose position numbers are relatively prime to 12—is:", "$$\n\boxed{40}\n$$"]