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.
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
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
The expression contains three main parts with parentheses:
- 3(x⁶ - 6x⁴ + 12x² - 8)
- -22(x⁴ - 4x² + 4)
- +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
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
Why Accurate Expansion Matters
Errors often creep in when expanding brackets—burned signs, missed coefficients, or mistake signs are common. Double-checking each step ensures correctness and strong foundational skills for advanced algebra, calculus, or engineering applications.
Conclusion
Simplifying 3(x⁶ - 6x⁴ + 12x² - 8) - 22(x⁴ - 4x² + 4) + 57x² - 166 (after combining constants) yields: 3x⁶ - 40x⁴ + 181x² - 278 With clear expansion, careful combination of like terms, and accurate arithmetic, polynomial simplification becomes manageable and reliable.
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