%2B(8x4%E2%88%923x3%2B4x%2B1).jpeg "(12x5−3x4+2x−5)+(8x4−3x3+4x+1)")
Simplifying Polynomial Expressions: A Step-by-Step Guide
This article will guide you through the process of simplifying the following polynomial expression:
(12x⁵−3x⁴+2x−5) + (8x⁴−3x³+4x+1)
Understanding the Basics
Before we dive into the simplification, let's refresh our understanding of key terms:
- Polynomial: An expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication.
- Term: A single part of a polynomial separated by addition or subtraction.
- Coefficient: The numerical factor of a term.
- Variable: A symbol that represents an unknown value.
- Degree: The highest power of the variable in a polynomial.
Simplifying the Expression
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Remove the parentheses: Since we're adding the two polynomials, the parentheses don't affect the order of operations. We can simply rewrite the expression without them:
12x⁵−3x⁴+2x−5 + 8x⁴−3x³+4x+1
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Combine like terms: Identify terms with the same variable and power. Add their coefficients while keeping the variable and power unchanged:
- x⁵ terms: 12x⁵
- x⁴ terms: -3x⁴ + 8x⁴ = 5x⁴
- x³ terms: -3x³
- x² terms: None
- x terms: 2x + 4x = 6x
- Constant terms: -5 + 1 = -4
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Write the simplified expression: Combine the simplified terms in descending order of their power:
12x⁵ + 5x⁴ - 3x³ + 6x - 4
Final Result
Therefore, the simplified form of the expression (12x⁵−3x⁴+2x−5) + (8x⁴−3x³+4x+1) is 12x⁵ + 5x⁴ - 3x³ + 6x - 4.