Problem 33 - Precalculus Mathematics for Calculus

Question

Multiply the algebraic expressions using a Special Product Formula and simplify. $$(2 x+3 y)^{2}$$

Step-by-Step Solution

Answer

The simplified expression is \(4x^2 + 12xy + 9y^2\).

Step 1: Identify the Formula

The given expression \((2x + 3y)^2\) is of the form \((a + b)^2\) where \(a = 2x\) and \(b = 3y\). This is a case for the special product formula for a square of a binomial: \((a+b)^2 = a^2 + 2ab + b^2\).

Step 2: Apply the Formula

Based on the identified values of \(a\) and \(b\), apply the special product formula: \((2x + 3y)^2 = (2x)^2 + 2(2x)(3y) + (3y)^2\).

Step 3: Calculate Each Term

Now, calculate each term individually:1. \((2x)^2 = 4x^2\)2. \(2(2x)(3y) = 12xy\)3. \((3y)^2 = 9y^2\)

Step 4: Combine the Terms

Combine the calculated terms from the previous step.The simplified expression is:\(4x^2 + 12xy + 9y^2\).

Key Concepts

Square of a Binomial

Understanding the square of a binomial is key to simplifying algebraic expressions with ease. When you encounter an expression like \((2x + 3y)^2\), it represents a binomial squared, which means multiplying the binomial by itself. The special product formula for the square of a binomial is given by:

• \((a + b)^2 = a^2 + 2ab + b^2\).

This formula tells us how to expand the squared binomial. Here, each term within the parentheses, such as \(a = 2x\) and \(b = 3y\), is used to find the expanded form.
By substituting these specific values into the formula, you can derive the expanded expression in a structured way. This helps in visualizing and solving algebraic problems, making it easier to grasp more complex polynomial multiplications.
Begin this process by identifying, then substituting, and finally simplifying using arithmetic operations.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operations that form the building blocks of algebra. In our example, \((2x + 3y)^2\) is an algebraic expression which consists of two terms:

• The first term, \(2x\), is a product of number 2 and variable \(x\).
• The second term, \(3y\), is a product of number 3 and variable \(y\).

These expressions are essential in algebra as they help in representing real-world situations in a mathematical form. You can simplify them, manipulate them through different operations, and combine them to solve equations.
Understanding how each part of an algebraic expression interacts is crucial. For example, knowing that \(2x\) and \(3y\) cannot be directly combined because they are unlike terms is important when simplifying an expression.
Using algebraic expressions efficiently can solve practical problems like calculating areas, optimizing operations, and analyzing trends.

Polynomial Multiplication

Multiplying polynomials may seem challenging at first, but using strategies like the special product formulas makes it simpler. For the expression \((2x + 3y)^2\), polynomial multiplication breaks down to:

• Squaring each term individually: \((2x)^2 = 4x^2\) and \((3y)^2 = 9y^2\).
• Calculating their product and multiplying by two: \[2(2x)(3y) = 12xy\].

After breaking down the expression, you recombine them to form the simplified polynomial: \(4x^2 + 12xy + 9y^2\). This approach ensures that every part of the binomial is correctly multiplied and added.
Polynomial multiplication is a foundational algebra skill. It provides a basis for learning more advanced mathematical concepts. Starting small with understanding each multiplication step leads to mastering more complex algebraic challenges.
Practice regularly with different polynomials to enhance your skills and build this essential knowledge over time. Working step-by-step, as shown, helps solidify the process in your learning journey.