["Understanding R(x) = (x - 1)²(x + 2): A Comprehensive Guide", "When exploring polynomial functions in algebra, expressions like R(x) = (x - 1)²(x + 2) offer valuable insight into the behavior of quadratics combined with linear factors. In this SEO-optimized article, we’ll break down what R(x) represents, how to expand and factor it, analyze its graph, and understand its real-world applications. Whether you're a student, educator, or autodidact, grasping this function helps strengthen your foundation in polynomial algebra.", "---", "### What is R(x) = (x - 1)²(x + 2)?", "R(x) = (x - 1)²(x + 2) is a cubic polynomial in standard form. It combines a squared quadratic factor and a linear factor, producing a function with key characteristics such as multiplicity of roots, turning points, and intercepts.", "---", "### Step-by-Step Expansion of R(x)", "Start with expanding the expression to identify its standard polynomial form:", "[
\nR(x) = (x - 1)²(x + 2) = (x² - 2x + 1)(x + 2)
\n]", "Now, perform the multiplication:", "[
\n= x²(x + 2) - 2x(x + 2) + 1(x + 2)
\n= x³ + 2x² - 2x² - 4x + x + 2
\n= x³ - 3x + 2
\n]", "So,
\nR(x) = x³ − 3x + 2", "Understanding this expanded form helps in graphing, calculus operations, and identifying critical points like maxima, minima, and inflection behaviors.", "---", "### Roots and Their Multiplicities", "From the original factored form R(x) = (x - 1)²(x + 2), we immediately identify the roots:", "- x = 1 with multiplicity 2 (a double root)
\n- x = -2 with multiplicity 1 (a simple root)", "This multiplicity influences the graph: at x = 1, the graph touches the x-axis and turns around (tangent behavior), while at x = -2, it crosses the x-axis normally.", "---", "### Graphing R(x): Shape and Key Features", "Since R(x) is cubic (degree 3), it has an S-shaped curve with:", "- y-intercept: R(0) = (0 – 1)²(0 + 2) = 1 × 2 = 2
\n- x-intercepts: at x = 1 (touches) and x = –2 (crosses)
\n- Turning points: A cubic polynomial can have up to two turning points; in this case, calculated via derivative.", "Derivative:
\nFrom R(x) = x³ − 3x + 2
\n[
\nR'(x) = 3x² - 3 = 3(x² - 1) = 3(x - 1)(x + 1)
\n]
\nSetting R’(x) = 0 gives critical points at x = –1 and x = 1.", "- At x = –1: local maximum or minimum?
\nUsing second derivative:
\n[
\nR''(x) = 6x \Rightarrow R''(-1) = -6 < 0 \Rightarrow \ ext{local maximum at } x = –1
\n]
\n- At x = 1: R''(1) = 6 > 0 ⇒ local minimum.", "Compute function values:
\n- R(–1) = (–1 – 1)²(–1 + 2) = (−2)²(1) = 4 × 1 = 4 (local max)
\n- R(1) = 0 (on x-axis)", "Thus, the graph rises to (–1, 4), dips to (1, 0), with a smooth turn at x = 1.", "---", "### Real-Life Applications of Cubic Polynomials Like R(x)", "Polynomial functions model real-world phenomena where change is non-linear but smooth. R(x) = (x – 1)²(x + 2) can represent:", "- Economic growth curves with inflection and saturation points
\n- Projectile motion with terminal height changes
\n- Volumetric relationships in physics where acceleration changes
\n- Profit functions crossing zero at break-even points with varying slopes", "Its shape reflects scenarios where initial change is rapid, then moderates (double root) before second-order decay (another double root? here singular—only one simple root)—but in this case, only one turnaround since multiplicity reduces influence.", "---", "### Summary: Why R(x) = (x – 1)²(x + 2) Matters", "- Expanded form: x³ − 3x + 2 — familiar standard cubic
\n- Roots with multiplicities: double root at x = 1, simple root at x = –2
\n- Graph features: local maximum at x = –1, minimum at x = 1, x-intercepts at –2 and 1
\n- Applications span: modeling dynamic systems with equilibrium or turning behavior", "---", "### Key SEO Keywords to Optimize This Content", "- R(x) = (x - 1)²(x + 2
\n- Factored polynomial R(x)
\n- Expanded form of cubic R(x)
\n- Graphing R(x) = (x - 1)²(x + 2
\n- Root multiplicities in cubic polynomials
\n- Real-world applications of cubic functions
\n- How to analyze turning points of R(x)
\n- Value of R(x) at x = 0, x = 1, x = –2
\n- Cubic polynomial shape and behavior", "---", "### Final Thoughts", "Understanding R(x) = (x – 1)²(x + 2) equips learners with fundamentals for analyzing cubic functions — critical in mathematics, engineering, economics, and natural sciences. Mastering expansion, root behavior, graphing, and context application paves the way for advanced study. Keep practicing with factoring, derivatives, and applications to fully harness the power of polynomials.", "---", "Meta Title:
\nR(x) = (x - 1)²(x + 2) | Complete Algebra Guide with Expansion & Graph", "Meta Description:
\nExplore R(x) = (x - 1)²(x + 2) — its expansion, roots, graph features, and real-world relevance. Learn how multiplicity shapes behavior in cubic polynomials. SEO-optimized for students and educators.", "---", "Keywords: R(x), cubic polynomial, factoring, graph R(x), polynomial roots, multiplicity, derivative analysis, real-world modeling, algebraic functions"]
