Simplifying Polynomial Expressions
This article will guide you through the process of simplifying the polynomial expression:
(x⁴  6x⁴y  x²y⁴)  (2x⁴y  2x²y⁴  6)
Understanding the Problem
The expression involves multiple terms with variables (x and y) raised to different powers. We need to simplify it by combining like terms.
Steps for Simplification

Distribute the negative sign: The minus sign in front of the second set of parentheses means we multiply each term inside the parentheses by 1:
(x⁴  6x⁴y  x²y⁴) + (2x⁴y + 2x²y⁴ + 6)

Identify like terms: Look for terms with the same variables raised to the same powers.
 x⁴ terms: x⁴ and 2x⁴y
 x⁴y terms: 6x⁴y
 x²y⁴ terms: x²y⁴ and 2x²y⁴
 Constant term: 6

Combine like terms: Add or subtract the coefficients of each set of like terms:
 x⁴ terms: x⁴  2x⁴y = x⁴  2x⁴y
 x⁴y terms: 6x⁴y = 6x⁴y
 x²y⁴ terms: x²y⁴ + 2x²y⁴ = x²y⁴
 Constant term: 6 = 6

Write the simplified expression: Combine the simplified terms:
x⁴  2x⁴y  6x⁴y + x²y⁴ + 6

Combine remaining like terms: Combine the x⁴y terms:
x⁴  8x⁴y + x²y⁴ + 6
Final Result
The simplified form of the polynomial expression (x⁴  6x⁴y  x²y⁴)  (2x⁴y  2x²y⁴  6) is x⁴  8x⁴y + x²y⁴ + 6.