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The binomial expansion involves the process of expanding expressions that are raised to a power. In the expression given in our exercise, we are expanding \((x-6)^{2}\). This can be understood through the concept of binomial expansion, which is essentially using a specific set of rules or formulas to simplify expressions that involve binomials, or two-term algebraic expressions.

The expansion of binomials can be illustrated using the binomial theorem, which provides a systematic way to expand expressions of the form \((a+b)^{n}\). In simpler terms, it shows how to multiply a binomial by itself multiple times, breaking it down to smaller, more manageable terms. For our specific example of \((x-6)^2\), while the binomial theorem technically applies to higher powers, the application of a more specialized formula such as the square of a binomial works as a shortcut. This saves time and reduces the likelihood of errors in computation.

When we talk about the square of a binomial, we are referring to the process of multiplying a two-term expression by itself. In mathematical terms, this is written as \((a-b)^2\), which expands to a pattern of products:

• \(a^2\) (the square of the first term)
• \(-2ab\) (twice the product of the two terms with a negative sign)
• \(b^2\) (the square of the second term)

For our example, the expression \((x-6)^2\) fits perfectly into this pattern.

Following the pattern, we substitute \(a\) with \(x\) and \(b\) with \(6\) to get \(x^2 - 2 \times x \times 6 + 6^2\). This simplifies further to give \(x^2 - 12x + 36\). Understanding this special product pattern helps simplify the process of squaring binomials and avoids lengthy multiplication.

Algebraic expressions are mathematical phrases that can include numbers, variables, and operation symbols. They don't always have to form a complete equation or equal something, but they often represent the terms and operations that need to be done in math problems. In this exercise, the expression \((x-6)^{2}\) is our main focus.

The process of dealing with algebraic expressions involves understanding the different parts and how they interact. Here, the expression is a binomial squared, and rearranging it using known formulas helps simplify it into a polynomial \((x^2 - 12x + 36)\). Each part of this expression needs to be calculated separately and then combined to give the final result. The key is to see how such expressions relate and use techniques like factoring or expansion to manage them efficiently.

Mastering algebraic expressions will not only help solve equations but will also streamline more complex mathematical problems by breaking them down into simpler components.