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When you hear the term "binomial expansion," it refers to the process of expanding expressions that involve binomials raised to a power. A binomial is any expression that contains two terms. For instance, in the given exercise, we have the binomial \(6z + \frac{1}{3}\).
When this binomial is squared, it means we are computing the product of the binomial with itself. The formula used for binomial expansion in this case is the square of a binomial formula:\[a^2 + 2ab + b^2\]Let's break it down:

• \(a\) represents the first term. In this example, \(a = 6z\).
• \(b\) stands for the second term, which is \(b = \frac{1}{3}\).

By plugging these into the formula, we set up the framework for the expansion and prepare the terms for further polynomial operations.

Polynomial operations, such as addition, subtraction, multiplication, and division, are crucial when handling algebraic expressions composed of several terms. In this problem, after using the binomial expansion formula, we have three terms: \((6z)^2\), \(2 \cdot 6z \cdot \frac{1}{3}\), and \((\frac{1}{3})^2\). Each term has a specific calculation:
1. Calculate \((6z)^2\): This involves squaring both the coefficient and the variable, resulting in \(36z^2\).
2. Evaluate \(2 \cdot 6z \cdot \frac{1}{3}\): First, multiply the coefficients \(2\), \(6\), and \(\frac{1}{3}\) together, simplifying to get \(4z\).
3. Compute \((\frac{1}{3})^2\): Square the fraction, resulting in \(\frac{1}{9}\).
These individual calculations are basic polynomial operations, which, when combined, formulate the expanded polynomial.

Simplifying expressions is a key step in solving algebraic problems and is all about combining like terms and minimizing the expression's complexity. After performing polynomial operations on the expanded binomial in our exercise, we obtain:

• \(36z^2\)
• \(4z\)
• \(\frac{1}{9}\)

Each term is distinct and does not contain like terms with others, meaning no further combination is possible. This results in a simplified expression:\[36z^2 + 4z + \frac{1}{9}\]This is the expression in its most reduced form, ready for any further algebraic manipulations or as the final answer to our problem. Simplifying makes working with algebraic expressions more manageable, especially in larger equations.