Problem 74 - Elementary Algebra

Question

The pattern for the difference of squares is given as \((a-b)(a+b)=a^{2}-b^{2}\). Is this equivalent to the pattern \((a+b)(a-b)=a^{2}-b^{2} ?\) Explain.

Step-by-Step Solution

Answer

Yes, \((a - b)(a + b) = (a + b)(a - b) = a^2 - b^2\).

Step 1: Identify the Given Pattern

The difference of squares pattern is given as \((a-b)(a+b) = a^2 - b^2\).

Step 2: Write Down the Alternate Pattern

The pattern to be compared is \((a+b)(a-b)\).

Step 3: Apply the Commutative Property of Multiplication

Recognize that multiplication is commutative, meaning that \((a + b)(a - b)\) is equal to \((a - b)(a + b)\). Hence, we can rearrange the terms without changing the result.

Step 4: Expand and Simplify Both Expressions

Expand both expressions. For the given pattern: \((a - b)(a + b)\) expands to \(a^2 - b^2\). For the alternate pattern: \((a + b)(a - b)\) also expands to \(a^2 - b^2\).

Step 5: Verify Equivalence

Since both expansions lead to the same result, \(a^2 - b^2\), it confirms that the patterns \((a - b)(a + b)\) and \((a + b)(a - b)\) are equivalent.

Key Concepts

Multiplicative Properties

Let's start with the **multiplicative properties**. These properties govern how multiplication works with numbers and variables. Two key multiplicative properties are particularly useful in algebra.

First, **the associative property** allows us to change the grouping of factors without changing the product. For example, \((a \times b) \times c = a \times (b \times c) \).

Second, **the distributive property** lets us multiply a single term across terms inside parentheses: \(a(b + c) = ab + ac \).

In the difference of squares pattern, we rely on these properties to manipulate and simplify expressions. Understanding these properties helps us see how and why expressions can be rearranged and still give the same result.

Commutative Property

Next, we discuss the **commutative property**. This property asserts that the order of numbers in multiplication doesn't affect the product. In other words, \(a \times b = b \times a \).

In the given exercise, the commutative property is crucial. It allows us to understand that \(a + b \times a - b\) is essentially the same as \(a - b \times a + b\). This reordering doesn't change the final product, shown as \(a^2 - b^2\).

Recognizing this property helps simplify complex algebraic expressions and confirms that two different-looking expressions can indeed be identical due to the commutative nature of multiplication.

Algebraic Patterns

**Algebraic patterns** are repeated sequences or structures in algebra that help solve problems efficiently. The difference of squares pattern is a vital algebraic pattern. It states that \(a^2 - b^2\) can be factored into \((a - b)(a + b)\).

By knowing this pattern, we can quickly factor quadratic expressions and solve equations. Recognizing algebraic patterns helps make complex equations more manageable and provides shortcuts for calculations.

These patterns are fundamental in algebra and beyond, forming the basis for more advanced mathematical concepts.

Equation Expansion

Finally, let's explore **equation expansion**. This term describes the process of multiplying out expressions to convert factored forms into polynomial forms.

In our example, we expanded \((a - b)(a + b)\) by using the distributive property:

• First: \a \times a = a^2\
• Outer: \a \times b = ab\
• Inner: \-b \times a = -ab\
• Last: \-b \times b = -b^2\

Adding these gives \(a^2 - b^2\) because the \+ab\ and \-ab\ terms cancel out.

Learning how to expand equations is vital for understanding algebra deeply. It enables us to manipulate and simplify expressions, making it easier to solve problems and understand their structure.