Problem 46 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (a-2 b)^{2} $$

Step-by-Step Solution

Answer

So, the product is \(a^{2} - 4ab + 4b^{2}\)

Step 1: Identify the binomial

Analyze the formula \((a-2 b)^{2}\). Here, \(a\) and \(2b\) are the components of the binomial.

Step 2: Apply the binomial square formula

Apply the formula for the square of a binomial: \((a-b)^{2}=a^{2}-2 a b+b^{2}\). Substitute \(a\) with \(a\) and \(b\) with \(2b\) to find \((a-2b)^{2}\).

Step 3: Calculate the square

\((a-2b)^{2} = a^{2}-2ab + (2b)^{2}\) \ After performing the calculations: \ \(= a^{2} - 4ab + 4b^{2}\)

Key Concepts

Simplifying Algebraic Expressions

Understanding how to simplify algebraic expressions is essential for working with equations and solving problems in algebra. Simplification involves reducing expressions to their simplest form without changing their value. To accomplish this, we combine like terms and use mathematical properties.

For instance, with the expression \( (a - 2b)^2 \) from our exercise, we simplify by identifying terms that can be combined or expanded. We recognize that a squared binomial can be expanded to a more simplified form utilizing the binomial square formula. Simplification isn't just about making an expression shorter; it's about preparing it for further operations such as solving or substituting variables. Remember to always combine like terms after expanding and to arrange terms in descending order of their degree for the standard form of a polynomial.

Polynomial Multiplication

Multiplying polynomials is a fundamental skill in algebra. It involves distributing each term of one polynomial with every term of another. This process is also referred to as the FOIL method when applied to binomials (First, Outer, Inner, Last).

In the case of squaring a binomial like \( (a-2b)^2 \), it translates to multiplying \( a-2b \) by itself. Rather than simplifying term by term, students expand the whole expression using the binomial square formula: \( a^2 - 2ab + b^2 \). This results in a trinomial, a type of polynomial with three terms. By meticulously applying polynomial multiplication, students ensure accuracy in simplifying algebraic expressions and solving further mathematical problems.

Exponent Rules

Exponent rules govern the operations involving powers in algebra. These rules make handling expressions with exponents more manageable. A critical principle used in our exercise is the Power of a Power rule, which states that when you raise an exponent to another exponent, you multiply the exponents: \( (b^n)^m = b^{n\times m} \).

When we have \( (a-2b)^2 \), we apply this rule to the term \( (2b)^2 \). Since the base is \( 2b \) and both the base and exponent have a power of 2, we apply the rule resulting in \( 4b^2 \). Understanding exponent rules is crucial for operations like polynomial multiplication because it ensures that terms are combined and simplified correctly, addressing the need for simplifying algebraic expressions accurately in a wider range of problems.