Problem 33 - Algebra and Trigonometry

Question

Perform the indicated operations and simplify. $$ (1-2 y)^{2} $$

Step-by-Step Solution

Answer

\((1-2y)^2 = 4y^2 - 4y + 1.\)

Step 1: Apply the Formula for Squaring a Binomial

The given expression is \[(1-2y)^2.\]We need to expand this using the formula for squaring a binomial which is \[(a-b)^2 = a^2 - 2ab + b^2.\]Here, \(a = 1\) and \(b = 2y\).

Step 2: Compute the Square of Each Term

Calculate \(a^2\), \(2ab\), and \(b^2\) using the values from Step 1.1. \(a^2 = 1^2 = 1\).2. \(2ab = 2 \times 1 \times 2y = 4y\).3. \(b^2 = (2y)^2 = 4y^2\).

Step 3: Substitute into the Binomial Formula

Substitute the values calculated in Step 2 into the squared binomial formula:\[(1-2y)^2 = 1 - 4y + 4y^2.\]

Step 4: Simplify the Expression

Arrange the terms in descending order of powers of \(y\):\[4y^2 - 4y + 1.\]This is the simplified form of the expression.

Key Concepts

Binomial Expansion

When dealing with algebra, especially when working with expressions like \((1-2y)^2\), the binomial expansion technique is crucial. Binomial expansion is a method that allows us to multiply out and simplify expressions raised to a power. The expression \((a-b)^2\) is a binomial, meaning it's an expression with two terms. The formula for expanding a binomial square is given by \((a-b)^2 = a^2 - 2ab + b^2\). This formula helps us express the expanded form using a clear structural pattern.
For instance, in our example \((1-2y)^2\), the expansion goes through a series of calculations:

• Identify \(a = 1\) and \(b = 2y\).
• Calculate \(a^2 = 1^2\), which is 1.
• Find \(2ab = 2 \times 1 \times 2y = 4y\).
• Compute \(b^2 = (2y)^2 = 4y^2\).

Combining these results gives us the expanded form: \(1 - 4y + 4y^2\). Binomial expansion simplifies the process, providing a consistent method for handling expressions like these.

Polynomial Simplification

After we expand a binomial expression, the next fundamental task is polynomial simplification. This process involves combining or arranging terms into their simplest form. For the binomial \((1-2y)^2\), once expanded to \(1 - 4y + 4y^2\), we begin simplifying.
The aim of simplification is to arrange the polynomial in a standard form, usually starting with the highest power of each term. The polynomial we derived consists of different power terms:

• Constant term: \(1\)
• Linear term: \(-4y\)
• Quadratic term: \(4y^2\)

To simplify, we arrange these in order of descending powers, yielding \(4y^2 - 4y + 1\). This ordered arrangement makes it easier to interpret and solve further algebraic problems, ensuring clarity and accuracy in mathematical computations.

Quadratic Expressions

Quadratic expressions are an integral part of algebra. They appear in various mathematical problems and equations. A quadratic expression generally has the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants. In our example, the expression \(4y^2 - 4y + 1\) is a quadratic since it has a term \(y^2\), which signifies the highest degree as 2.
Working with quadratic expressions involves recognizing their structure:

• Quadratic term: \(4y^2\) dictates the expression's degree and shape.
• Linear term: \(-4y\) influences the slope and position.
• Constant term: \(1\) shifts the graph vertically.

Quadratics can be factored, solved, or used to create graphs representing parabolas. Understanding how to expand, simplify, and arrange them is crucial because these skills form the foundation for solving many real-world problems in physics, engineering, and economics. Mastering quadratics enables you to approach these problems with confidence.