Simplifying Polynomial Expressions: A StepbyStep 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 nonnegative 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

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.