Problem 45 - College Algebra

Question

For the following exercises, find the sum or difference. $$(a+b)(a-b)$$

Step-by-Step Solution

Answer

The sum or difference simplifies to \(a^2 - b^2\).

Step 1: Identify the Formula

Recognize the formula used in the given expression \((a+b)(a-b)\). This expression is an application of the difference of squares formula: \((a+b)(a-b) = a^2 - b^2\).

Step 2: Apply the Difference of Squares

Apply the difference of squares formula to the expression \((a+b)(a-b)\). This formula simplifies the product into a difference: \[ (a+b)(a-b) = a^2 - b^2 \]

Step 3: Calculate Each Term

Expand the expression by calculating each term in the equation. First calculate \(a^2\), which is \(a\) multiplied by itself, and then calculate \(b^2\), which is \(b\) multiplied by itself.

Step 4: Write the Simplified Expression

Combine the terms you found in the previous step into a simplified expression: \[ a^2 - b^2 \]

Key Concepts

Algebraic Expressions

Algebraic expressions are like mathematical sentences that involve numbers, variables, and operations like addition, subtraction, multiplication, and division. In the exercise, we are working with the expression \((a+b)(a-b)\). Here, \(a\) and \(b\) are variables, and we are multiplying two binomials. Binomials are expressions containing two terms, in this case, connected by addition or subtraction.

Understanding algebraic expressions is crucial. It allows us to express and solve various math problems. Here's why they matter:

• They provide a compact way to represent mathematical situations.
• They help us generalize mathematical relationships.
• They support the formulation of equations and inequalities.

Knowing how to manipulate algebraic expressions leads to deeper mathematical capabilities, such as simplifying expressions and solving equations.

Polynomial Identities

Polynomial identities are equations that are true for all values of the variables involved. They provide shortcuts for calculations and help simplify complex algebraic expressions. An example of a polynomial identity is the difference of squares identity, which states that \((a+b)(a-b) = a^2 - b^2\).

The beauty of polynomial identities lies in their ability to simplify tedious computations. The difference of squares is a versatile identity used to simplify expressions and solve equations.

Polynomial identities like this:

• Help recognize patterns in algebraic expressions.
• Provide efficient ways to factor and expand polynomials.
• Enable quick computation and can reveal relationships between variables.

In practice, these identities save time and reduce errors by allowing a systematic approach to simplifying expressions.

Mathematical Formulas

Mathematical formulas are predefined equations that describe a rule or relationship between numbers and variables. In this context, the difference of squares formula \((a+b)(a-b) = a^2 - b^2\) provides a rule for quickly expanding or simplifying the product of two conjugate binomials.

Formulas like this one play a pivotal role in mathematics. Here’s why they’re important:

• They provide quick solutions to complex problems.
• They encapsulate common patterns and relationships.
• They allow abstraction, letting us solve for unknowns systematically.

By using mathematical formulas, we can engage with math problems more efficiently, identifying forms and applying rules that streamline computations.