Problem 86 - Elementary Algebra

Question

We can write the difference of squares pattern as \(a^{2}-b^{2}=(a+b)(a-b)\) or as \(a^{2}-b^{2}=\) \((a-b)(a+b)\). Show that \((a+b)(a-b)=\) \((a-b)(a+b)\).

Step-by-Step Solution

Answer

(a+b)(a-b) = (a-b)(a+b) due to the commutative property of multiplication.

Step 1 - Recall the Difference of Squares

The difference of squares formula states that the difference between the squares of two numbers, a and b, can be expressed as the product of their sum and difference:\[ a^2 - b^2 = (a+b)(a-b) \].

Step 2 - Analyze the Expression

Note that the expression we need to verify is \((a+b)(a-b)\) and \((a-b)(a+b)\).

Step 3 - Apply the Commutative Property

By applying the commutative property of multiplication, which states that the order of factors does not affect the product, we can see that \((a+b)(a-b) = (a-b)(a+b)\). This is because multiplication is commutative.

Key Concepts

Difference of Squares formula

The Difference of Squares formula is a powerful algebraic tool that helps simplify and solve a variety of mathematical problems. The formula is expressed as:

• \( a^2 - b^2 = (a+b)(a-b) \)

This tells us that the difference between the squares of two numbers, \( a^2 \) and \( b^2 \), can be factored into the product of their sum \( (a+b) \) and their difference \( (a-b) \). To illustrate why this works, let's expand the expression \( (a+b)(a-b) \):

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

Notice how the middle terms \( -ab \) and \( +ab \) cancel each other out, leaving us with \( a^2 - b^2 \). This confirms the formula.

Commutative Property of Multiplication

The commutative property of multiplication states that changing the order of the factors does not change the product. In other words, \( a \times b = b \times a \). This property is essential in algebra because it allows us to rearrange and simplify expressions. In the context of the Difference of Squares formula, the commutative property helps us understand why \( (a+b)(a-b) = (a-b)(a+b) \)

• Both expressions represent the same set of factors: \( (a+b) \) and \( (a-b) \).
• Due to the commutative property, the order of multiplication does not matter.
• Therefore, \( (a+b)(a-b) \) is equal to \( (a-b)(a+b) \).

This symmetry is why both forms of the Difference of Squares formula are valid and interchangeable.

Algebraic expressions

Algebraic expressions are combinations of variables, constants, and operators (such as +, -, ×, ÷). They are foundational elements in algebra that represent mathematical relationships and operations. Here are some key facts:

• Difference of squares is one type of algebraic expression.
• Expressions can often be simplified or factored using known formulas and properties.
• In the formula \( a^2 - b^2 \), \( a \) and \( b \) are variables or constants, while the operators are subtraction and exponentiation.

Factoring expressions like the difference of squares is a critical skill because it helps solve equations and simplify complex expressions. Understanding these fundamentals of algebraic expressions makes the study of algebra more approachable and less daunting.