Problem 22 - Algebra and Trigonometry with Analytic Geometry

Question

Express as a polynomial. $$ (5 x+4 y)(5 x-4 y) $$

Step-by-Step Solution

Answer

The polynomial is \(25x^2 - 16y^2\).

Step 1: Identify the Binomials

We are given the product of two binomials: \[(5x + 4y)(5x - 4y)\]These binomials are in the form of \((a + b)(a - b)\).

Step 2: Recall the Formula for Difference of Squares

The product of conjugate binomials \((a + b)(a - b)\) is a difference of squares, given by:\[a^2 - b^2\]

Step 3: Apply the Formula

In our binomials, \(a = 5x\) and \(b = 4y\). Apply the difference of squares formula:\[(5x)^2 - (4y)^2\]

Step 4: Simplify the Expression

Calculate each square:- \((5x)^2 = 25x^2\)- \((4y)^2 = 16y^2\)Thus, the expression becomes:\[25x^2 - 16y^2\]

Step 5: Final Expression as a Polynomial

The simplified expression is already a polynomial:\[25x^2 - 16y^2\]

Key Concepts

Binomials

Binomials are algebraic expressions that contain exactly two terms. These terms can consist of numbers, variables, or both. For example, in the expression \((5x + 4y)\), the two terms are \(5x\) and \(4y\). Recognizing binomials is essential in algebra because they form the basis of many operations, such as multiplication and factoring. Here are some key points about binomials:

• A binomial is a polynomial with two terms.
• Each term in a binomial can be a combination of numbers and variables.
• Common operations with binomials include addition, subtraction, and multiplication.
• Multiplying binomials often involves the use of special patterns, such as the difference of squares.

By understanding how to recognize and manipulate binomials, students can solve problems more effectively and apply these skills to higher-level math concepts. They are a building block for understanding more complex expressions and polynomials.

Difference of Squares

The "difference of squares" is a mathematical concept that makes it easier to simplify certain algebraic expressions. Specifically, when you have a product of two binomials that are conjugates, such as \((a + b)(a - b)\), the result is a difference of squares. This simplified expression is written as \(a^2 - b^2\).Understanding the difference of squares can help you quickly solve expressions without lengthy calculations. Here's what you need to know:

• The formula for the difference of squares: \((a + b)(a - b) = a^2 - b^2\).
• This formula works with any numbers or variables where there is a sum and a difference.
• The difference of squares can simplify problems, reduce errors, and save time.

By recognizing and applying the difference of squares formula, students can efficiently approach polynomial expressions that might seem complicated at first glance. This is a key skill in algebra and paves the way for more advanced topics in mathematics.

Simplifying Expressions

Simplifying expressions is a crucial step in solving algebraic problems. It involves reducing an expression to its simplest form, making it easier to understand and work with. In our example, once we apply the difference of squares to the binomials \((5x + 4y)(5x - 4y)\), we arrive at the simplified expression \(25x^2 - 16y^2\).Here are some tips for simplifying expressions:

• Apply known algebraic identities, like the difference of squares, to simplify calculations.
• Focus on each component of the expression, like individual square terms, to break down the problem.
• Combine like terms if possible, to further reduce the expression efficiently.
• Always double-check your calculations to ensure accuracy in the simplified expression.

Simplifying expressions not only helps in understanding a problem better but also makes subsequent calculations or evaluations more straightforward. Mastery of simplification can be a powerful tool, helping students navigate both basic and complex mathematical problems with confidence and clarity.