So $ R(x) = (x - 1)(x^2 + x - 2) $. Factor the quadratic:
["SEO Article: Factoring the Quadratic in the Expression $ R(x) = (x - 1)(x^2 + x - 2) $", "Understanding how to factor polynomial expressions is a fundamental skill in algebra, essential for solving equations, analyzing functions, and simplifying complex algebraic forms. Today, we explore the expression $ R(x) = (x - 1)(x^2 + x - 2) $ and provide a detailed breakdown of factoring the quadratic component — a key step toward fully simplifying the function.", "---", "### What is $ R(x) = (x - 1)(x^2 + x - 2) $?", "This expression combines two parts: a linear factor $ (x - 1) $ and a quadratic trinomial $ x^2 + x - 2 $. Factoring the full expression involves expanding and then factoring the quadratic, unlocking deeper insight into the behavior and roots of the function.", "---", "### Step 1: Factor the Quadratic $ x^2 + x - 2 $", "The quadratic $ x^2 + x - 2 $ cannot be factored easily by inspection, so we apply the standard method of factoring trinomials with leading coefficient 1.", "We seek two numbers $ m $ and $ n $ such that:\n$$\nm \cdot n = -2 \quad \ ext{and} \quad m + n = 1\n$$", "After testing the integer factor pairs of $ -2 $ — namely $ (2, -1), (-2, 1), (1, -2), (-1, 2) $ — we find that $ 2 $ and $ -1 $ satisfy both conditions:", "$$\n2 \cdot (-1) = -2 \quad \ ext{and} \quad 2 + (-1) = 1\n$$", "Hence, the quadratic factors as:\n$$\nx^2 + x - 2 = (x + 2)(x - 1)\n$$", "---", "### Step 2: Substitute Back into $ R(x) $", "Now substitute the factored quadratic into the original expression:\n$$\nR(x) = (x - 1)(x^2 + x - 2) = (x - 1)(x + 2)(x - 1)\n$$", "Since $ (x - 1) $ appears twice, we can write this as:\n$$\nR(x) = (x - 1)^2(x + 2)\n$$", "---", "### Why Factor This Quadratic?", "Factoring $ x^2 + x - 2 $ simplifies $ R(x) $ into irreducible components, revealing the roots and behavior of the function:", "- The roots are $ x = 1 $ (with multiplicity 2) and $ x = -2 $ (multiplicity 1), found by setting $ R(x) = 0 $:\n$$\n(x - 1)^2(x + 2) = 0 \Rightarrow x = 1,\ x = -2\n$$", "- Knowing this helps graph the function, analyze its symmetry, and solve inequalities efficiently.", "---", "### Final Factored Form and Conclusion", "The fully factored expression is:\n$$\nR(x) = (x - 1)^2(x + 2)\n$$", "By breaking down the quadratic via factoring, we transform the original product into a form that clarifies the algebraic structure. Whether you're solving equations, analyzing function behavior, or teaching foundational algebra, mastering this technique enhances mathematical fluency.", "For additional practice factoring quadratics, explore our guides on completing the square, quotient rule factoring, and trigonometric identities that simplify polynomial forms.", "---", "Keywords: factor $ R(x) $, quadratic factoring, factor $ x^2 + x - 2 $, simplify algebraic expressions, factor polynomial, algebra tutorial, roots of polynomials, irreducible quadratic, multiplicity in factoring", "Meta Description:\nLearn how to factor $ R(x) = (x - 1)(x^2 + x - 2) $ by breaking down the quadratic into $ (x + 2)(x - 1) $. Unlock simpler forms, find roots, and master essential algebraic techniques.", "---", "Incorporating systematic factoring transforms complex expressions into insightful, usable forms. Start mastering this today!"]
