Problem 41 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (a+2 b)(a-2 b) $$

Step-by-Step Solution

Answer

The product of \((a+2b)(a-2b)\) is \(a^2 - 4b^2\).

Step 1: Identify difference of squares

The product \((a+2b)(a-2b)\) is in the form \((a+b)(a-b)\) which can be expressed as the difference of two squares \(a^2 - b^2\). Here, \(a\) remains the same, while \(b\) is \(2b\)

Step 2: Apply difference of squares

Apply the difference of two squares formula. Replace \(a\) by \(a\) and \(b\) by \(2b\) in the formula. It gives the product as \(a^2 - (2b)^2\).

Step 3: Calculate the product

Simplify the expression to get the product. \(a^2 - (2b)^2\) equals \(a^2 - 4b^2\).

Key Concepts

Polynomial Expressions

A polynomial expression is a mathematical phrase made up of variables, coefficients, and the operations of addition, subtraction, and multiplication. In our original exercise, the expression \((a+2b)(a-2b)\) is a product of two binomials. Polynomials are classified based on the number of terms they have. A binomial, like those in the expression, is a polynomial with exactly two terms. It involves each term having a variable raised to a whole number exponent, and these are combined using the operations mentioned.
Understanding polynomial expressions is important because they form the foundation of algebra and help in solving equations and inequalities. Here are some key characteristics of polynomial expressions:

• They consist only of terms with non-negative integer exponents.
• They do not contain variables in denominators or under root symbols.
• The highest exponent of a variable in the expression is called the degree.

Recognizing the structure of polynomial expressions, like seeing that \((a+2b)(a-2b)\) fits the difference of squares pattern, allows for easier simplification and manipulation of algebraic forms.

Multiplication of Binomials

The multiplication of binomials is a fundamental algebraic skill, used to expand expressions like \((a+2b)(a-2b)\). One common method to multiply binomials is to apply the FOIL method, which stands for First, Outer, Inner, Last. Although in the case of difference of squares, there's a shortcut we can take.
Let's break it down without shortcuts first:

• First: Multiply the first terms in each binomial, producing \(a \times a = a^2\).
• Outer: Multiply the outer terms, producing \(a \times (-2b) = -2ab\).
• Inner: Multiply the inner terms, producing \(2b \times a = 2ab\).
• Last: Multiply the last terms, producing \(2b \times (-2b) = -4b^2\).

Now, simplify by combining like terms: the \(-2ab + 2ab\) cancels out, leaving us with \(a^2 - 4b^2\), a perfect example of the difference of squares. Taking note of special patterns, like difference of squares, helps simplify such expressions quickly without fully expanding every time.

Simplification of Algebraic Expressions

Simplifying algebraic expressions involves reducing the expression to its simplest form while maintaining its equivalence. In the context of the given problem, simplification allows us to transform the expression \(a^2 - (2b)^2\) into \(a^2 - 4b^2\). This step requires applying arithmetic and recognizing structure within the expressions.
For simplification:

• Identify like terms or obvious arithmetic operations: Here, \((2b)^2\) simplifies to \(4b^2\).
• Apply known algebraic identities like difference of squares to make expressions manageable.
• Eliminate redundant terms that cancel out, ensuring the expression is as concise as possible.

Being comfortable with simplification means being able to, quickly and efficiently, rewrite expressions in their simplest form, often saving time and preventing errors in more complex algebraic manipulations.