Problem 52 - Algebra and Trigonometry

Question

Multiply the algebraic expressions using a Special Product Formula, and simplify. \((1-2 y)^{2}\)

Step-by-Step Solution

Answer

The simplified expression is \(4y^2 - 4y + 1\).

Step 1: Identify the Special Product Formula

The problem involves squaring a binomial expression, specifically \((1 - 2y)^2\). This corresponds to the algebraic identity \((a - b)^2 = a^2 - 2ab + b^2\). In this expression, \(a = 1\) and \(b = 2y\).

Step 2: Apply the Special Product Formula

Using the formula \((a - b)^2 = a^2 - 2ab + b^2\), substitute \(a = 1\) and \(b = 2y\). The expression becomes \((1)^2 - 2(1)(2y) + (2y)^2\).

Step 3: Calculate Each Term Separately

First, calculate \(a^2\): \((1)^2 = 1\). Next, calculate \(-2ab\): \(-2(1)(2y) = -4y\). Finally, calculate \(b^2\): \((2y)^2 = 4y^2\).

Step 4: Combine the Calculated Terms

Combine the calculated terms: \(1 - 4y + 4y^2\). Make sure each term is simplified.

Key Concepts

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and operations. They are the building blocks of algebra. Think of them as recipes with ingredients like numbers, letters (representing variables), and various operations such as addition, subtraction, multiplication, and division.
In exercises like the one presented, you are often asked to manipulate these expressions to arrive at a simplified or desired form. Recognizing and working with algebraic expressions is crucial for grasping more complex mathematical concepts.

• An algebraic expression could look as simple as \(2x + 3\) or as complex as \((1 - 2y)^2\).
• Expressions can be simplified or multiplied using various techniques, such as the special product formulas.
• Key components of these expressions include constants (like 1 or 3) and variables (like \(x\) and \(y\)).

Exploring Binomial Squaring

Squaring a binomial refers to multiplying a binomial by itself. A binomial is a simple algebraic expression containing two terms connected by a plus or minus sign. The expression you are working with, \((1 - 2y)^2\), involves squaring a binomial.
Using the formula for squaring binomials,

• The squared binomial formula is \((a - b)^2 = a^2 - 2ab + b^2\).
• In this case, \(a = 1\) and \(b = 2y\).
• You substitute these values into the formula to expand the binomial into three terms.

This approach simplifies the process and ensures accuracy because it avoids mistakenly multiplying incorrectly.

The Art of Simplification

Simplification is the process of making an expression easier to understand or solve. When you simplify, you want to combine like terms and ensure each component is reduced to its simplest form.
In the given problem, after expanding the expression using the binomial squaring formula, you end up with \(1 - 4y + 4y^2\). Simplifying algebraic expressions requires careful calculation of each term before combining them.

• First, calculate \((1)^2\) to get 1.
• Next, compute \(-2(1)(2y)\) to obtain \(-4y\).
• Lastly, evaluate \((2y)^2\) to reach \(4y^2\).

Bring all these results together to form the simplified expression.

The Power of Algebraic Identities

Algebraic identities such as the special product formulas are powerful tools in mathematics. They tell us that certain patterns or structures are always true for all values of the variables involved. They simplify the process of expanding and simplifying expressions.
For example, knowing the identity \((a - b)^2 = a^2 - 2ab + b^2\) allows you to predict the outcome when squaring any binomial in the form of \((a - b)\). These identities save time and reduce mistakes when managing algebraic expressions.

• They help you quickly and accurately expand expressions like \((1 - 2y)^2\).
• They provide a consistent framework for solving complex algebraic problems.
• Understanding these identities is essential for advancing in algebra and mathematics generally.

Mastering algebraic identities equips you with the tools to tackle a wide range of mathematical challenges with confidence.