Thus, $ g(x^2 - 2) = 3(x^2 - 2)^3 - 22(x^2 - 2)^2 + 57(x^2 - 2) - 52 $. Expanding:
["Title: Expanding $ g(x^2 - 2) = 3(x^2 - 2)^3 - 22(x^2 - 2)^2 + 57(x^2 - 2) - 52 $: Step-by-Step Expansion Explained", "Understanding how to expand complex expressions like $ g(x^2 - 2) $ is crucial in algebra, calculus, and applied mathematics. This article walks you through the step-by-step expansion of the function\n$$\ng(x^2 - 2) = 3(x^2 - 2)^3 - 22(x^2 - 2)^2 + 57(x^2 - 2) - 52\n$$\nwith a clear breakdown of each transformation. Whether you're preparing for a math exam, developing a function model, or simply sharpening your algebraic skills, this guide will help you master the expansion process.", "---", "### Understanding the Function $ g $", "The expression defines a function $ g(u) $ where $ u = x^2 - 2 $. That is:\n$$\ng(u) = 3u^3 - 22u^2 + 57u - 52\n$$\nOur task is to substitute $ u = x^2 - 2 $ into this polynomial and expand fully.", "---", "### Step 1: Substitute $ u = x^2 - 2 $ into $ g(u) $", "Replace every instance of $ u $ with $ x^2 - 2 $:\n$$\ng(x^2 - 2) = 3(x^2 - 2)^3 - 22(x^2 - 2)^2 + 57(x^2 - 2) - 52\n$$", "---", "### Step 2: Expand $ (x^2 - 2)^3 $", "Use the binomial expansion:\n$$\n(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\n$$\nLet $ a = x^2 $, $ b = 2 $:\n$$\n(x^2 - 2)^3 = (x^2)^3 - 3(x^2)^2 \cdot 2 + 3(x^2)(2^2) - 2^3 = x^6 - 6x^4 + 12x^2 - 8\n$$", "Multiply by 3:\n$$\n3(x^2 - 2)^3 = 3x^6 - 18x^4 + 36x^2 - 24\n$$", "---", "### Step 3: Expand $ (x^2 - 2)^2 $", "$$\n(x^2 - 2)^2 = x^4 - 4x^2 + 4\n$$\nMultiply by -22:\n$$\n-22(x^2 - 2)^2 = -22x^4 + 88x^2 - 88\n$$", "---", "### Step 4: Expand $ 57(x^2 - 2) $", "$$\n57(x^2 - 2) = 57x^2 - 114\n$$", "---", "### Step 5: Combine all terms", "Now add all expanded parts:\n$$\n\begin{align}\ng(x^2 - 2) &= (3x^6 - 18x^4 + 36x^2 - 24) \\n&\quad + (-22x^4 + 88x^2 - 88) \\n&\quad + (57x^2 - 114) \\n&\quad - 52\n\end{align}\n$$", "Group like terms:", "- $ x^6 $ term: $ 3x^6 $\n- $ x^4 $ terms: $ -18x^4 - 22x^4 = -40x^4 $\n- $ x^2 $ terms: $ 36x^2 + 88x^2 + 57x^2 = 181x^2 $\n- Constant terms: $ -24 - 88 - 114 - 52 = -278 $", "---", "### Final Expanded Polynomial", "$$\ng(x^2 - 2) = 3x^6 - 40x^4 + 181x^2 - 278\n$$", "---", "### Why This Expansion Matters", "- Function Analysis: Expressing $ g $ in terms of $ x^2 $ reveals symmetry and helps identify transformations if $ g $ is a general polynomial.\n- Graphing: Expanded form simplifies plotting and analyzing the behavior of $ g(x^2 - 2) $ as a function of $ x $.\n- Solving Equations: Useful when solving $ g(x^2 - 2) = k $ for real $ x $, as polynomial equations are easier to manage in expanded form.\n- Calculus Applications: Facilitates differentiation and integration via a clearer algebraic expression.", "---", "### Summary", "Expanding $ g(x^2 - 2) = 3(x^2 - 2)^3 - 22(x^2 - 2)^2 + 57(x^2 - 2) - 52 $ results in:\n$$\n\boxed{3x^6 - 40x^4 + 181x^2 - 278}\n$$\nThis step-by-step expansion combines power expansions, coefficient distribution, and careful term combination to yield a clean, usable polynomial form. Mastering this process enables deeper insight into functional transformations and algebraic manipulation.", "---", "Keywords: expand $ g(x^2 - 2) $, algebraic expansion, polynomial expansion, binomial theorem, function substitution, calculus prep, step-by-step math, equation solving, intermediate algebra.", "---", "Start expanding with confidence—each power, coefficient, and term builds the foundation for advanced mathematics."]
