91Ó°ÊÓ

Algebraic expressions are a combination of variables, constants, and operations such as addition, subtraction, multiplication, and division.
They are essential components in algebra that represent quantities that can change. For example, in the expression \(21 b^{6} - 36 b^{4}\), 21 and -36 are constants, while \(b\) is a variable raised to specific powers of 6 and 4. Understanding how to manipulate these expressions is key to solving algebraic equations.

In any algebraic expression, terms are separated by addition or subtraction signs. Each term includes variables raised to a power and may also include numbers (coefficients). Understanding how these terms relate through various operations is the foundation of algebra.
This problem demonstrates algebraic manipulations involving both coefficients and exponents, essential skills for dividing polynomials effectively.

Dividing polynomials is an extension of the basic division operation applied to algebraic expressions containing variables. The goal is to simplify the expression by dividing each term in the polynomial by the divisor.

In this exercise, the dividend is \(21 b^{6} - 36 b^{4}\) and the divisor is \(3 b^{3}\). To perform the division, each term in the dividend is divided separately by the divisor. This involves dividing both the coefficients and simplifying the powers of the variables:

• Divide coefficients: Take \(21\) and \(-36\) and divide by \(3\). This gives the new coefficients \(7\) and \(-12\).
• Subtract exponents: For the variable \(b\), subtract the exponent of the divisor \(b^{3}\) from each term’s power in the dividend. Thus, \(b^{6-3} = b^3\) and \(b^{4-3} = b^1\), simplified to \(b\).

Hence, dividing polynomials involves these key steps of breaking down each term and reducing the expression.

Simplifying expressions is a crucial step in dealing with algebraic equations, making them easier to understand or solve. A simplified expression means all operations like addition, subtraction, multiplication, and division have been completed as much as possible.

In this context, after dividing the polynomial, we obtain two terms \(7b^3\) and \(-12b\). The simplified expression \(7b^3 - 12b\) is the result of the division and represents the simplest form of the original expression \((21 b^{6} - 36 b^{4}) \div (3 b^{3})\).

Through simplification, we reduced the complexity of the polynomial, leaving us with fewer terms and ensuring that all like terms have been combined and unnecessary operations have been removed. This practice of simplifying is crucial in algebra as it allows focusing on the core elements of the equation without extra baggage, essential for solving and understanding mathematical problems.