(6x-7x^2+7)-(5x^2+2x-2x^3-1)

3 min read Jun 16, 2024
(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

  1. 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
    
  2. Combine like terms: Identify and group terms with the same variable and exponent.

    2x^3  - 7x^2 - 5x^2 + 6x - 2x + 7 + 1
    
  3. 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.

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