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When you encounter two binomials that you need to multiply, the process involves using the distributive property, often referred to as FOIL (First, Outer, Inner, Last). This means you multiply the first terms, the outer terms, the inner terms, and then the last terms of each binomial before combining like terms. For example, when we have ewline ewline ewline (\(x - 1\))^2, which is the same as ewline ewline ewline \((x - 1)(x - 1)\), we apply FOIL to get: ewline ewline ewline First: \(x \times x = x^2\), ewline ewline ewline Outer: \(x \times -1=-x\), ewline ewline ewline Inner: \(-1 \times x=-x\), ewline ewline ewline Last: \(-1 \times -1=1\). ewline ewline ewline Then, we combine like terms (the middle terms in this case) to conclude with \(x^2 - 2x + 1\).

In the realm of algebra, expressions represent a combination of numbers, variables, and mathematical operations. For instance, the expression ewline ewline ewline \((x-1)^2\) is algebraic as it includes the variable \(x\), the constant \(-1\), an exponent, and implied multiplication between \(x-1\) and itself. To work with algebraic expressions effectively, it's essential to understand the order of operations and the properties of algebra that allow us to manipulate expressions into simpler or more useful forms.

Simplifying algebraic expressions is the process of making them as basic as possible. This involves combining like terms, which are terms that have the same variables raised to the same powers. In the solution for ewline ewline ewline (\(x-1\))^2, after we expanded the expression by FOIL, we combine the like terms \(-x\) and \(-x\) which simplifies the expression to ewline ewline ewline \(x^2 - 2x + 1\). Simplification is an important step since it enables us to compare and manipulate expressions more easily and can be key in solving algebraic equations.

The binomial theorem is a fundamental principle in algebra that provides a shortcut to expand expressions raised to a power without multiplying each term individually. This is incredibly handy for large powers. It tells us that to expand ewline ewline ewline (\(a+b\))^n, where \(n\) is a non-negative integer, we can express it as a sum involving coefficients known as binomial coefficients. These coefficients can be found on Pascal's Triangle or by using the combination formula ewline ewline ewline \(C(n,k) = \frac{n!}{k!(n-k)!}\). For the case of ewline ewline ewline (\(x-1\))^2, which is to say ewline ewline ewline \(n = 2\), this is a straightforward application of the theorem resulting in a squared term, a single product term, and a constant term, ultimately simplifying to ewline ewline ewline \(x^2 - 2x + 1\).