Problem 47 - Elementary and Intermediate Algebra

Question

Find each special product. $$ (5 b-2)^{2} $$

Step-by-Step Solution

Answer

The expanded form is \(25b^2 - 20b + 4\).

Step 1: Recognize the Pattern

The expression \((5b - 2)^2\) is a binomial squared. We utilize the formula \((a-b)^2 = a^2 - 2ab + b^2\), where \(a = 5b\) and \(b = 2\). This formula simplifies the squared term.

Step 2: Apply the Formula for Squaring a Binomial

Substitute \(a = 5b\) and \(b = 2\) into the formula \((a-b)^2 = a^2 - 2ab + b^2\). This gives us:\[(5b - 2)^2 = (5b)^2 - 2(5b)(2) + (2)^2\]

Step 3: Calculate Each Term Separately

Now, calculate each part of the expression:1. \((5b)^2 = 25b^2\)2. \(-2(5b)(2) = -20b\)3. \((2)^2 = 4\)

Step 4: Combine the Calculated Terms

Combine the results from Step 3:\[25b^2 - 20b + 4\]

Key Concepts

Understanding Special Products

Special products are a notable category in algebra where certain products of binomials have predictable results. Understanding these can simplify computations and problem-solving in algebraic contexts. The key special products formulas include the square of a binomial, the product of binomials, and the difference of squares. These formulas are efficient shortcuts. Instead of expanding expressions manually, you can directly apply the formulas to find the product.
The exercise in question involves recognizing the special product formula used for a squared binomial. Identifying special products is crucial because:

• They save time by skipping lengthy multiplication or FOIL steps.
• They simplify calculations, especially when dealing with complex algebraic expressions.

Special products not only simplify the arithmetic but also enhance your understanding of how numbers and variables interact within mathematical expressions. This fundamental grasp over special products lays the groundwork for advanced algebraic manipulations and problem-solving skills.

Squaring a Binomial

Squaring a binomial involves finding the square of an expression of the form \((a-b)^2\) or \((a+b)^2\). It entails a specific algebraic expansion resulting from multiplying the binomial by itself. These formulas help:

• \((a+b)^2 = a^2 + 2ab + b^2\)
• \((a-b)^2 = a^2 - 2ab + b^2\)

Understanding squaring a binomial is paramount as it allows us to transform an expression like \((5b - 2)^2\) into its expanded form \(25b^2 - 20b + 4\). The process involves three simple steps:

• Squaring the first term: \((5b)^2 = 25b^2\)
• Doubling the product of both terms: \(-2(5b)(2) = -20b\)
• Squaring the second term: \((2)^2 = 4\)

Combining these, the original expression is expanded effortlessly, converting it into a sum of terms: a polynomial. This conversion is a critical skill for handling quadratic equations and polynomial functions later in math studies.

Exploring Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. They represent generalized instructions for how to calculate numbers. When working with algebraic expressions, particularly in contexts such as the special product, understanding the components is essential.
In our example, the binomial \((5b - 2)\) is an algebraic expression made up of:

• A variable term \(5b\), which involves a coefficient and a variable.
• A constant term \(-2\).

Algebraic expressions are foundational in algebra as they symbolize real-life quantities and operations, allowing flexible manipulation through operations like addition, subtraction, multiplication, and division. Recognizing variable and constant components of these expressions paves the way for deeper mathematical exploration, simplifying problems that would otherwise require repetitive calculations.
Being able to expand, factor, and simplify these expressions is a lifelong mathematical tool, giving you power and flexibility in both pure math and applied contexts, such as physics and engineering.