Problem 45 - College Algebra with Corequisite Support

Question

For the following exercises, multiply the polynomials. \((a+b)(a-b)\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Pattern

Notice that the expression \((a+b)(a-b)\) follows the pattern of the difference of squares, which is \((x+y)(x-y) = x^2 - y^2\). In this case, \(x = a\) and \(y = b\).

Step 2: Apply the Formula

Use the difference of squares formula \((a+b)(a-b) = a^2 - b^2\) to obtain the product of the two binomials.

Step 3: Simplify the Expression

Calculate \(a^2\) and \(b^2\). The expression can be directly written as \(a^2 - b^2\).

Key Concepts

Difference of Squares

When you encounter an expression like \((a+b)(a-b)\), you are dealing with a well-known algebraic identity called the difference of squares. The difference of squares occurs when you multiply two binomials that are in the form \((x+y)(x-y)\). This pattern is special because it simplifies directly into a simple expression without any middle terms.
The general formula is:

• \((x+y)(x-y) = x^2 - y^2\)

This identity works because when you expand \((x+y)(x-y)\), the middle terms cancel out:

• \((x+y)(x-y)\)
• = \(x^2 - xy + xy - y^2\)
• = \(x^2 - y^2\)

In the exercise, setting \(x = a\) and \(y = b\), we directly apply:

• \((a+b)(a-b) = a^2 - b^2\)

So, whenever you notice this pattern, you can easily simplify the expression using this identity. Understanding this concept makes multiplying certain binomials much faster and helps recognize patterns in larger polynomial expressions, which is a common occurrence in algebra.

Binomials

Binomials are a type of algebraic expression that contains exactly two distinct terms. For example, both \(a+b\) and \(a-b\) are binomials because they include two terms each.
Binomials can be identified easily by looking for two terms separated by either a plus or minus sign. They play a vital role in algebra, especially when multiplying or factoring expressions. Understanding how to work with binomials makes it easier to handle polynomial equations.
Some common operations with binomials include:

• Addition: Combine like terms.
• Subtraction: Distribute the negative sign and combine like terms.
• Multiplication: Use distributive property or special identities such as the difference of squares.

In the given exercise, you multiply two binomials \((a+b)\) and \((a-b)\). By recognizing this as a difference of squares, you simplify the multiplication process significantly. You don't need to think about each term individually; instead, you use the identity to arrive at \(a^2 - b^2\). Binomials show up frequently in algebra, and mastering them is crucial to progressing with more complex mathematical problems.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition, subtraction, multiplication, and division) prepared in a meaningful way. These expressions form the backbone of algebra because they represent either specific values or more abstract relationships.
Characteristics of algebraic expressions include:

• Composed of terms, where each term is a product of numbers and variables.
• Can be classified based on the number of terms: monomials (one), binomials (two), trinomials (three), polynomials (multiple).
• Operations of expressions involve simplifying, factoring, and solving them.

In the exercise involving \((a+b)(a-b)\), the two binomials are multiplied to create a new expression. Understanding how to transform these expressions through multiplication into a simpler form, like in this case \(a^2 - b^2\), is part of manipulating algebraic expressions.
Recognizing patterns, such as the difference of squares, helps bypass lengthy multiplication steps. Plus, it allows you to convert expressions into more manageable forms for further computations or solving equations. Therefore, recognizing and transforming algebraic expressions with efficiency is a valuable skill in algebra.