Problem 39 - Algebra 1: Concepts and Skills

Question

Find the product. $$ (2 y+5)(2 y-5) $$

Step-by-Step Solution

Answer

The product of the given binomials is \(4y^2 - 25\).

Step 1: Identify the terms

In the binomials, you can see that \(a = 2y\) and \(b = 5\).

Step 2: Apply the formula

Now, apply the difference of squares formula \(a^2 - b^2\) which gives us \((2y)^2 - (5)^2).\)

Step 3: Simplify the result

Simplify the square of each term to get the final answer. \(2y * 2y = 4y^2\) and \(5 * 5 = 25\). Therefore, the product is \(4y^2 - 25\).

Key Concepts

Understanding Binomial Products

When dealing with binomial products, it's important to first understand what a binomial is. A binomial is simply an algebraic expression that contains two terms, usually connected by a plus or minus sign. In the exercise you've seen, both expressions - \(2y+5\) and \(2y-5\) - are binomials.

Multiplying binomials often requires using specific formulas, such as the difference of squares. The difference of squares is a common pattern in algebra. It states that for any two numbers or expressions \(a\) and \(b\), the product \((a+b)(a-b)\) is equal to \(a^2 - b^2\). This formula helps simplify the multiplication of two binomials that are conjugates, meaning they have the same terms but different middle signs.

In your exercise, you have \(a = 2y\) and \(b = 5\). By recognizing the difference of squares, you can quickly find the product without expanding each term individually. This not only saves time but also helps in getting more accurate results in problems involving algebraic expressions.

Simplifying Expressions with Formulas

Simplifying expressions means breaking down or rewriting an algebraic expression to make it easier to work with. This is a crucial skill in algebra and higher-level mathematics, as it is the basis for solving equations and understanding complex functions.

When using the difference of squares formula, each term in the expression is squared and then subtracted. In this case:

• Square the first term: \((2y)^2 = 4y^2\)
• Square the second term: \((5)^2 = 25\)

Then, perform the subtraction \(4y^2 - 25\).

This process not only simplifies the problem but also provides a clear and concise solution. Simplifying expressions is essential for finding solutions and for communicating mathematical ideas effectively.

Exploring Algebraic Expressions

Algebraic expressions represent values that can change and are comprised of constants, variables, and operations. A key part of working with algebra is learning to manipulate and simplify these expressions to make them useful for solving equations.

Expressions like \(2y\) and \(5\) are examples of what you can encounter. The variable \(y\) can take different values, which is why understanding how to handle algebraic expressions is vital.

By multiplying, simplifying, or rearranging terms, you can solve equations or simplify them to a form that's more understandable. In the example \((2y+5)(2y-5)\), the goal is to simplify into a form using basic algebraic rules, like the difference of squares. This allows us to easily evaluate or further manipulate expressions in various mathematical contexts.

Understanding these processes is fundamental to mastering algebra and enhances your ability to tackle more complex mathematical problems later on.