Problem 79 - Algebra and Trigonometry

Question

In Exercises 67–82, find each product. $$ (3 x+5 y)(3 x-5 y) $$

Step-by-Step Solution

Answer

The product of the given binomials is \(9x^2 - 25y^2\).

Step 1: Identify \(a\) and \(b\)

In the given expression, \(a\) is represented by \(3x\) and \(b\) is represented by \(5y\). The expression can be written as a multiplication of two binomials of the form \(a+b\) and \(a-b\).

Step 2: Apply the Difference of Squares Formula

You can immediately apply the formula \(a^2 - b^2 = (a+b)(a-b)\). Replacing \(a\) with \(3x\) and \(b\) with \(5y\), you get: \[(3x)^2 - (5y)^2\]. This formulation of the problem allows for a clear calculation without multiplicative expansion.

Step 3: Simplify the Result

Calculate each of the squares separately: \( (3x)^2 = 9x^2\) and \( (5y)^2 = 25y^2\). Then subtract the two resulting squares, giving you \(9x^2 - 25y^2\).

Key Concepts

Multiplying Binomials

The process of multiplying binomials is an essential skill in algebra, often resulting in a specific pattern that students can learn to recognize. When multiplying two binomial expressions, for example, \((x + y)(x - y)\), we are in fact distributing each term of the first binomial across the second binomial.

This action is following the FOIL (First, Outer, Inner, Last) method. It's a way of ensuring that each part of the binomials is multiplied together. However, in the case of the difference of squares, the inner and outer terms cancel each other out because one is positive and the other is negative, but both have the same absolute value. Therefore, multiplying the binomials \((3x + 5y)\) and \((3x - 5y)\) simplifies by removing the inner and outer terms to just focus on the first and last. Simplified, it becomes \(a^2 - b^2\), skipping the need for the full FOIL process.

Algebraic Expressions

An algebraic expression is a mathematical phrase that includes numbers, variables, and operations symbols. Examples include \(2x + 3y\) or \(a^2 - b^2\). Such expressions are the bread and butter of algebra and serve as the building blocks for forming equations and solving complex problems.

These expressions can be as simple as a single term or as complicated as a polynomial with many terms. The expression \((3x + 5y)(3x - 5y)\), while appearing complex, is an algebraic expression that can be simplified and understood through the use of algebraic rules and formulas. Recognizing the structure of these expressions makes a variety of advanced operations, like the difference of squares, more approachable.

Factoring Polynomials

The factoring polynomials technique is a reverse process of multiplying polynomials. In factoring, we are trying to break down a complex expression into simpler, multipliable factors. If we look at the formula \(a^2 - b^2 = (a+b)(a-b)\), we can understand it as the process of factoring a difference of two squares.

The product \((3x + 5y)(3x - 5y)\) provides a classic example of this, where the simplified form \(9x^2 - 25y^2\) is the factored result of the original binomials. Learning to factor polynomials like this allows for solving quadratic equations, simplifying expressions, and understanding the fundamental structure of algebraic equations.