91Ó°ÊÓ

The distributive property is a fundamental concept in algebra that helps in expanding or simplifying expressions. It states that for any numbers or variables, the product of a number and a sum (or difference) inside the parentheses can be distributed to each term inside the parentheses. So, for example:

\( a(b + c) = ab + ac \)

Applied to subtraction, this means: \( a(b - c) = ab - ac \)

In the given problem, we distribute the negative sign through the second polynomial before subtracting:
\( -(3 - 6y^3 - 8y^2) = -3 + 6y^3 + 8y^2 \)
This changes the signs of each term within the parentheses, making it easier to combine like terms later.

Combining like terms is a process where terms that have the same variable and exponent are added or subtracted together. It simplifies the expression for easier computation. For instance, terms like \( 2x^2 \) and \( -5x^2 \) can be combined because they both have \( x^2 \).

After distributing the negative sign in the exercise, we can rewrite the polynomial without parentheses:
\( 8y^2 + y - 11 - 3 + 6y^3 + 8y^2 \)
Next, we gather all the like terms together:
\( 6y^3 \, 8y^2 + 8y^2 \, y \, -11 - 3 \)

We then combine these to get:
\( 6y^3 + 16y^2 + y - 14 \)

Polynomials are algebraic expressions that consist of variables and coefficients, structured in terms with non-negative integer exponents. Each term in a polynomial forms part of the overall expression. For example: \( ax^n + bx^{n-1} + ... + zx^0 \).

Basic operations with polynomials include addition, subtraction, multiplication, and sometimes division. Understanding how to work with polynomials is crucial for solving various algebraic problems.

In this exercise, we dealt with the polynomials:
\ (8y^2 + y - 11) \ and \ (3 - 6y^3 - 8y^2) \
Simplifying these by subtraction and combining the like terms resulted in:
\ 6y^3 + 16y^2 + y - 14 \

This demonstrates how to manage polynomials through basic algebraic operations.