Problem 26 - Elementary Algebra

Question

Factor completely. $$ 25 x 2-9 y 2 $$

Step-by-Step Solution

Answer

The complete factorization of the expression is \((5x - 3y)(5x + 3y)\).

Step 1: Recognize the Expression Type

The expression given is a subtraction of two squares: \(25x^2 - 9y^2\). This is a difference of squares because \(25x^2\) is \((5x)^2\), and \(9y^2\) is \((3y)^2\).

Step 2: Apply the Difference of Squares Formula

The difference of squares can be factored using the formula: \((a^2 - b^2) = (a - b)(a + b)\). Here, \(a = 5x\) and \(b = 3y\).

Step 3: Write the Factored Form

Substitute \(a = 5x\) and \(b = 3y\) into the difference of squares formula: \((5x)^2 - (3y)^2\) becomes \((5x - 3y)(5x + 3y)\).

Key Concepts

Difference of Squares

In algebra, the difference of squares is a very specific type of factoring for expressions that can be written in the form \(a^2 - b^2\). This particular pattern is known because each term is a perfect square and they are subtracted from each other. The unique property of the difference of squares is that it can always be factored using the formula \((a^2 - b^2) = (a - b)(a + b)\).
This formula stems from the basic arithmetic property that multiplying these conjugates results in the squares difference: \((a-b)(a+b) = a^2 - b^2\).

• This means to factor a difference of squares, you determine what squares make up each part of the expression.
• In our example, \(25x^2 - 9y^2\), it's clear that \(25x^2 = (5x)^2\) and \(9y^2 = (3y)^2\). This makes \(a = 5x\) and \(b = 3y\).

With these values, using the difference of squares formula is straightforward and turns an initially complicated expression into an easily manageable product.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. They are used to represent real-world situations or complex mathematical ideas. In our example, \(25x^2 - 9y^2\), we see an algebraic expression that includes both a numerical coefficient and variables raised to a power.
Understanding algebraic expressions involves identifying and working with these components:

• Coefficients: The numerical part, like 25 and 9, which multiplies the variable parts.
• Variables: These are symbols, like \(x\) and \(y\), representing quantities that can change.
• Exponents: Indicate how many times the variable is multiplied by itself, such as 2 in \(x^2\).
• Operations: Including addition, subtraction, multiplication, etc., to combine the parts.

Recognizing the structure of an algebraic expression allows you to apply necessary mathematical techniques for simplification or, like in this case, for factoring.

Factoring Formulas

Factoring is a method used to break down expressions into their simplest building blocks, called factors, that when multiplied, recreate the original expression. Factoring formulas are key tools in this process, providing straightforward means for common types of expressions.
For the expression \(25x^2 - 9y^2\), we use the difference of squares methodology. However, it's one of several factoring formulas students might encounter:

• Difference of Squares: \((a^2 - b^2) = (a-b)(a+b)\)
• Perfect Square Trinomials: \((a+b)^2 = a^2 + 2ab + b^2\)
• Sum of Cubes: \((a^3 + b^3) = (a+b)(a^2 - ab + b^2)\)
• Difference of Cubes: \((a^3 - b^3) = (a-b)(a^2 + ab + b^2)\)

While only the difference of squares is needed for our current example, understanding and recognizing these formulas can help simplify a wide range of algebraic expressions. Mastery of these formulas can make working with complex expressions more manageable, opening the door to advanced algebraic manipulation.