Problem 83 - Algebra and Trigonometry

Question

In Exercises 83–90, perform the indicated operation or operations.. $$ (3 x+4 y)^{2}-(3 x-4 y)^{2} $$

Step-by-Step Solution

Answer

The simplified form of the given expression is \( 48xy \)

Step 1: Expanding the squares

First, apply the formula for the square of a binomial. The square of a binomial \( (a+b)^2 \) is given by \( a^2 + 2ab + b^2 \). The square of a binomial \( (a-b)^2 \) is given by \( a^2 - 2ab + b^2 \). Applying these to the given expression, we get \[ (3x + 4y)^2 = (3x)^2 + 2(3x)(4y) + (4y)^2 = 9x^2 + 24xy + 16y^2 \] \[ (3x - 4y)^2 = (3x)^2 - 2(3x)(4y) + (4y)^2 = 9x^2 - 24xy + 16y^2 \] Therefore, the original expression becomes \[ 9x^2 + 24xy + 16y^2 - (9x^2 - 24xy + 16y^2) \]

Step 2: Subtracting the expressions

Now, perform the subtraction by distributing the negative sign into the parentheses: \[ 9x^2 + 24xy + 16y^2 - 9x^2 + 24xy - 16y^2 \]

Step 3: Simplifying the expression

Finally, combine like terms: \[ (9x^2 - 9x^2) + (24xy + 24xy) + (16y^2 - 16y^2) = 48xy \]

Key Concepts

Binomial Expansion

The binomial expansion is a method used to expand expressions that are raised to a power, specifically expressions of the form \((a+b)^n\). When you have a binomial raised to a power, it's important to expand it using a recognized pattern, usually Pascal's Triangle or binomial theorem. However, in simpler cases like squaring, you just need to use formulas:- The formula for the square of a sum: \((a+b)^2 = a^2 + 2ab + b^2\).- The formula for the square of a difference: \((a-b)^2 = a^2 - 2ab + b^2\).In the given exercise, both binomials \((3x+4y)^2\) and \((3x-4y)^2\) were expanded using these rules. This method helps to break down and understand each component of the expansion, making it easier to handle more complex expressions.

Simplifying Expressions

Simplifying expressions involves performing arithmetic operations and combining like terms to represent an expression in its simplest form. The goal is to make the expression as compact and straightforward as possible.- First, apply negative signs carefully, especially when subtracting entire expressions.- Identify like terms, which usually share the same variables raised to the same powers, and combine them.In our exercise, after expanding both squared terms, subtraction was necessary. This required distributing the negative sign across the terms of the second binomial expansion:- Change the signs of the terms in the second expression: \(- (9x^2 - 24xy + 16y^2) = - 9x^2 + 24xy - 16y^2\).- Combine like terms from the expanded binomials into a simple term.This results in a significant reduction in complexity, simplifying a potentially confusing problem.

Polynomials

Polynomials are expressions that include constants, variables, and exponents, combined using addition, subtraction, and multiplication. They're fundamental in algebra and appear in many forms.- A polynomial can have multiple terms, each consisting of a product of a number (coefficient) and variables raised to an exponent.- When dealing with polynomials, order terms from the highest power of the variables to the lowest for clarity.In the exercise, both expansions \((9x^2 + 24xy + 16y^2)\) and \((9x^2 - 24xy + 16y^2)\) are polynomials. We also see how they simplify through subtraction to a simpler polynomial \(48xy\).Understanding how to manipulate and simplify polynomials helps solve not only algebraic but also real-world problems, making clear insights obtainable from complex situations.