Simplifying Polynomial Expressions: (6x2x^3+1)(7x^254x^3)
This article will guide you through the process of simplifying the polynomial expression: (6x2x^3+1)(7x^254x^3). We'll break down each step to ensure a clear understanding.
Understanding the Basics
Before we start simplifying, let's define some key terms:
 Polynomial: An expression consisting of variables and constants, combined using addition, subtraction, and multiplication, where the variables have nonnegative integer exponents.
 Term: A single part of a polynomial separated by addition or subtraction.
 Coefficient: The numerical factor in front of a variable.
 Like Terms: Terms with the same variables raised to the same exponents.
Simplifying the Expression

Distribute the Negative Sign: The minus sign in front of the parentheses means we multiply each term inside the second set of parentheses by 1:
(6x  2x^3 + 1) + (1)(7x^2  5  4x^3)
This gives us:
6x  2x^3 + 1  7x^2 + 5 + 4x^3

Combine Like Terms: Now we identify terms with the same variables and exponents:
 x^3 terms: 2x^3 + 4x^3 = 2x^3
 x^2 terms: 7x^2
 x terms: 6x
 Constant terms: 1 + 5 = 6

Write the Simplified Expression: Combining the like terms, we get:
2x^3  7x^2 + 6x + 6
Final Result
The simplified form of the expression (6x2x^3+1)(7x^254x^3) is 2x^3  7x^2 + 6x + 6.
By following the steps outlined above, you can confidently simplify any polynomial expression involving addition, subtraction, and parentheses.