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Simplifying Polynomials: (x² + 1 - 4x³ + 3x⁴) + (-x² - 2x⁴ + x³ + 4)
This article explores the process of simplifying the given polynomial expression: (x² + 1 - 4x³ + 3x⁴) + (-x² - 2x⁴ + x³ + 4). We will break down the steps involved in combining like terms and arriving at the simplified form.
Understanding Polynomials
A polynomial is an expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication. The degree of a polynomial is determined by the highest exponent of the variable. In our expression, the highest exponent is 4, making it a fourth-degree polynomial.
Combining Like Terms
To simplify the expression, we need to combine like terms. Like terms are terms with the same variable and exponent. Let's group the terms by their variable and exponent:
- x⁴ terms: 3x⁴ - 2x⁴
- x³ terms: -4x³ + x³
- x² terms: x² - x²
- Constant terms: 1 + 4
Performing the Operations
Now, we can combine the coefficients of each group:
- x⁴ terms: (3 - 2)x⁴ = x⁴
- x³ terms: (-4 + 1)x³ = -3x³
- x² terms: (1 - 1)x² = 0
- Constant terms: 1 + 4 = 5
The Simplified Expression
Finally, we combine the results to obtain the simplified polynomial:
x⁴ - 3x³ + 5
Therefore, the simplified form of the expression (x² + 1 - 4x³ + 3x⁴) + (-x² - 2x⁴ + x³ + 4) is x⁴ - 3x³ + 5.