-(5x%5E2%2B2x-2x%5E3-1).jpeg "(6x-7x^2+7)-(5x^2+2x-2x^3-1)")
Simplifying Polynomial Expressions: A Step-by-Step Guide
This article will guide you through the process of simplifying the polynomial expression (6x - 7x^2 + 7) - (5x^2 + 2x - 2x^3 - 1).
Understanding the Basics
Before we begin, let's define some key terms:
- Polynomial: A mathematical expression consisting of variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents.
- Terms: Individual parts of a polynomial separated by addition or subtraction signs.
- Coefficient: The numerical factor that multiplies a variable in a term.
- Like terms: Terms that have the same variables raised to the same exponents.
Simplifying the Expression
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Distribute the negative sign: The minus sign in front of the second parenthesis means we multiply each term inside the parenthesis by -1.
(6x - 7x^2 + 7) - (5x^2 + 2x - 2x^3 - 1) = 6x - 7x^2 + 7 - 5x^2 - 2x + 2x^3 + 1 -
Combine like terms: Identify and group terms with the same variable and exponent.
2x^3 - 7x^2 - 5x^2 + 6x - 2x + 7 + 1 -
Simplify: Combine the coefficients of like terms.
2x^3 - 12x^2 + 4x + 8
Final Result
The simplified form of the polynomial expression (6x - 7x^2 + 7) - (5x^2 + 2x - 2x^3 - 1) is 2x^3 - 12x^2 + 4x + 8.
Remember:
- When adding or subtracting polynomials, we only combine like terms.
- Always pay attention to the signs, especially when distributing a negative sign.
- Simplify the expression by combining coefficients of like terms.
By following these steps, you can successfully simplify any polynomial expression.