Simplifying 3(x⁶ - 6x⁴ + 12x² - 8) - 22(x⁴ - 4x² + 4) + 57x² - 114 - 52: A Step-by-Step Simplification

Simplifying polynomial expressions can often feel like solving a puzzle—especially when nested parentheses, coefficients, and multiple terms are involved. In this article, we’ll break down the complex expression:
3(x⁶ - 6x⁴ + 12x² - 8) - 22(x⁴ - 4x² + 4) + 57x² - 114 - 52
and simplify it step by step to arrive at the final form:
3x⁶ - 40x⁴ + 181x² - 278

Understanding the Context

Why Simplifying Matters

Working with polynomials is essential across mathematics, engineering, computer science, and data analysis. Simplifying expressions reduces errors, improves readability, and makes it easier to analyze behavior—critical when solving equations, plotting graphs, or optimizing functions.

Step 1: Expand Each Bracketed Term

3(x^6 - 6x^4 + 12x^2 - 8) - 22(x^4 - 4x^2 + 4) + 57x^2 - 114 - 52 = 3x^6 - 18x^4 + 36x^2 - 24 - 22x^4 + 88x^2 - 88 + 57x^2 - 166 = 3x^6 - 40x^4 + 181x^2 - 278.

Key Insights

The expression contains three main parts with parentheses:

  1. 3(x⁶ - 6x⁴ + 12x² - 8)
  2. -22(x⁴ - 4x² + 4)
  3. +57x² - 114 - 52 (constant terms)

To simplify, we start by expanding the first two polynomial brackets.

Expanding 3(x⁶ - 6x⁴ + 12x² - 8)

Multiply each term inside the parentheses by 3:
= 3·x⁶ - 18x⁴ + 36x² - 24
= 3x⁶ - 18x⁴ + 36x² - 24

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Final Thoughts

Expanding -22(x⁴ - 4x² + 4)

Multiply each term by -22:
= -22x⁴ + 88x² - 88

Step 2: Combine All Expanded Terms

Now replace the original grouped expressions with their expanded forms:
Original expression becomes:
(3x⁶ - 18x⁴ + 36x² - 24) + (-22x⁴ + 88x² - 88) + 57x² - 114 - 52

Combine like terms:

  • x⁶ term: only 3x⁶ (no other x⁶ terms)
  • x⁴ terms: -18x⁴ - 22x⁴ = -40x⁴
  • x² terms: 36x² + 88x² + 57x² = (36 + 88 + 57)x² = 181x²
  • Constant terms: -24 - 88 - 114 - 52

Calculate constants:
-24 - 88 = -112
-112 - 114 = -226
-226 - 52 = -278

Final Simplified Expression

Putting it all together:
3x⁶ - 40x⁴ + 181x² - 278