A computational linguist uses a cubic model \( p(u) = u^3 - 3u^2 + 2u \) to analyze syntactic complexity. Determine the number of distinct real roots of \( p(u) = 0 \).

["Using a Cubic Model to Analyze Syntactic Complexity: The Roots of ( p(u) = u^3 - 3u^2 + 2u )", "In computational linguistics, quantifying syntactic complexity is essential for understanding language structure and readability. One powerful approach involves modeling syntactic behavior using mathematical functions. A cubic polynomial, due to its flexibility and richness in root behavior, is frequently applied in linguistic analysis. Here, we examine the cubic function:", "[\np(u) = u^3 - 3u^2 + 2u\n]", "### Understanding the Polynomial", "To analyze syntactic complexity, we seek the number of distinct real roots of ( p(u) = 0 ). Roots of a polynomial correspond to critical thresholds where syntactic patterns shift—useful in automatically assessing sentence structure.", "We begin by factoring ( p(u) ):", "[\np(u) = u(u^2 - 3u + 2)\n]", "Now factor the quadratic:", "[\nu^2 - 3u + 2 = (u - 1)(u - 2)\n]", "Thus, the full factorization is:", "[\np(u) = u(u - 1)(u - 2)\n]", "### Finding the Roots", "Setting ( p(u) = 0 ), we solve:", "[\nu(u - 1)(u - 2) = 0\n]", "The solutions are:", "[\nu = 0, \quad u = 1, \quad u = 2\n]", "All three roots are real and distinct.", "### Visual Interpretation: Cubic Shape and Real Roots", "The function ( p(u) ) is a cubic polynomial with leading coefficient positive (( +1 )), so its graph rises to ( +\infty ) as ( u \ o \infty ) and falls to ( -\infty ) as ( u \ o -\infty ). With three real roots separated by sign changes, the function crosses the horizontal axis exactly three times—confirming the three distinct real roots visually and algebraically.", "### Implications for Syntactic Complexity", "In computational linguistics, such cubic models can map semantic density or syntactic embedding depth. Each real root marks a qualitative change in linguistic structure—such as a shift from simple to compound constructions. With three distinct real roots, the model identifies three key syntactic complexity thresholds, offering quantifiable benchmarks for parsing algorithms and readability scoring.", "### Conclusion", "The cubic function ( p(u) = u^3 - 3u^2 + 2u ) has exactly three distinct real roots: ( u = 0 ), ( u = 1 ), and ( u = 2 ). These roots provide a clear mathematical foundation for identifying critical points in syntactic structure, demonstrating the power of computational linguistics in transforming linguistic patterns into analyzable models.", "By leveraging such models, researchers and language technology developers gain precise tools to measure and interpret syntactic complexity—bridging linguistics, mathematics, and artificial intelligence."]