Problem 30 - Precalculus Mathematics for Calculus

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 given expression is \((1-2y)^2\), which is a perfect square binomial. This matches the special product formula \((a-b)^2 = a^2 - 2ab + b^2\). Here, let \(a = 1\) and \(b = 2y\).

Step 2: Apply the 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

Calculate each term separately:- \((1)^2 = 1\)- \(-2(1)(2y) = -4y\)- \((2y)^2 = 4y^2\)

Step 4: Simplify the Expression

Combine all the terms obtained from Step 3:\[1 - 4y + 4y^2\]Reorder them in standard form:\[4y^2 - 4y + 1\]

Key Concepts

Perfect Square Binomials

A perfect square binomial is an algebraic expression that can be represented as the square of a binomial. Binomials are expressions containing two terms, often connected by a plus or minus sign. The perfect square binomial formula is a special product formula that specifically handles the square of a binomial.
The formula is expressed as:

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

To use this formula, identify \(a\) and \(b\) in the binomial \((a-b)\). Then, apply the formula to square the binomial. This simplifies complex multiplication into separate, more manageable smaller parts. The resulting expression from using this formula consists of three terms. Understanding these formulas can greatly simplify handling of algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations like addition, subtraction, multiplication, and division. They form the foundation for much of algebra.Spaces within these expressions are filled by variables, representing unknowns or quantities that can vary.

• Variables: Symbols such as \(x\), \(y\), or \(z\) are commonly used to stand in for unknown numbers.
• Constants: Fixed values like \(1\), \(2\), or any specific number.
• Terms: The separate elements that make up an expression, usually connected by addition or subtraction.

A binomial is a specific type of algebraic expression that includes exactly two terms.For example, \(1 - 2y\) is a binomial with constant \(1\) and a term \(-2y\). Understanding these components helps in applying techniques like the special product formulas.

Polynomial Simplification

Polynomial simplification involves expressing a polynomial in its simplest form by performing operations and combining like terms. Polynomials are types of algebraic expressions made up of multiple terms, often represented in standard form, where terms are ordered by decreasing powers.The process usually involves:

• Combining like terms: Terms that have the same variable raised to the same power, such as combining \(4y^2\) with another \(y^2\) term.
• Ordering terms: Arranging terms from highest degree to lowest degree, like changing \(1 - 4y + 4y^2\) to \(4y^2 - 4y + 1\).
• Simplifying calculations: Performing multiplication or division as indicated.

Using special product formulas during this process helps in not only simplifying each term but also ensuring the overall expression is orderly and easy to understand. The goal is to make the expression as efficient as possible, with all like terms neatly combined and ordered.