Problem 101 - Algebra and Trigonometry

Question

Explaining the Concepts. Explain how to square a binomial difference. Give an example with your explanation.

Step-by-Step Solution

Answer

To square a binomial difference, use the formula \( (a - b)^2 = a^2 - 2ab + b^2 \). For instance, for (x - 3)^2, 'a' is 'x' and 'b' is '3'. The square resolves to \( (x)^2 - 2(x)(3) + (3)^2 \), which simplifies to \( x^2 - 6x + 9 \)

Step 1: Understanding the Formula

Before anything else, it is critical to understand the formula for squaring a binomial difference. Simply put, the square of a binomial difference \( (a-b)^2 \) expands to \( a^2 - 2ab + b^2 \). This formula is a guide that allows us to expand the expression correctly. The square of a binomial is not simply squaring each term. Instead, the formula incorporates a middle term, -2ab, which is derived from applying the distributive property twice.

Step 2: Applying the Formula

After understanding the formula, the next step is to apply this to a specific example. Choose an example - let's use (x - 3)^2 as an example. Here, 'a' corresponds to 'x' and 'b' corresponds to '3'.

Step 3: Performing the Operation

Use the formula to expand (x - 3)^2. Replace 'a' with 'x' and 'b' with '3'. Applied to the formula, the operation would work out as follows: \( (x)^2 - 2(x)(3) + (3)^2 \)

Step 4: Simplify the Result

The operation in step 3 provides a result that can be simplified further. \( (x)^2 - 2(x)(3) + (3)^2 \) simplifies to \( x^2 - 6x + 9 \)

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like x or y), and operators (such as plus, minus, multiplication, and division). When dealing with algebraic expressions, it's important to understand how to manipulate these phrases to simplify or solve them.

For example, when squaring a binomial difference—a specific type of algebraic expression—you're effectively multiplying the binomial by itself. It's crucial not to make the common mistake of only squaring the individual terms. Instead, consider the interplay between the terms using the distributive property, which will introduce an additional middle term. In the expression \(a - b)^2\), 'a' and 'b' could represent any numbers or variables, and squaring the binomial difference requires careful expansion to capture all parts of the expression.

Binomial Theorem

Understanding the Power of Squaring Binomials

The binomial theorem provides a systematic way to expand expressions that raise a binomial to a power. The binomial theorem states that \( (a + b)^n \) can be expanded, and the coefficients of the resulting terms are determined by Pascal's triangle or the binomial coefficients. However, when squaring a binomial, specifically when \(n\) is 2, the pattern simplifies to \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\) for binomial sums and differences, respectively.

These expressions follow the same pattern as dictated by the theorem, which helps students understand and memorize the proper structure of the expansion without having to rely on repeated distribution. It is the foundation for correctly expanding binomials raised to any power and is particularly helpful for understanding the nature of quadratic expressions.

Distributive Property

Apply It Twice and Simplify

The distributive property is a cornerstone of algebra and critical to understanding how to square a binomial difference. It states that in the expression \(a(b + c)\), you can distribute \(a\) to both \(b\) and \(c\), resulting in \(ab + ac\). When squaring a binomial difference, like \(a - b)^2\), the distributive property must be applied twice.

First, you distribute the 'a' across the binomial \(a(a - b)\), yielding \(a^2 - ab\), and then distribute the '-b' likewise, resulting in \-ab + b^2\). Combining these using addition, remembering that \-ab\ and \-ab\ are like terms, produces the middle term of -2ab, yielding the complete squared expression \(a^2 - 2ab + b^2\). As shown in our example with \(x - 3)^2\, the distributive property confirms the importance of including the middle term, which could be overlooked if one was to incorrectly only square the individual terms of the binomial.