Problem 5 - Algebra for College Students

Question

Use the difference-of-squares pattern to factor each of the following. $$9 x^{2}-25 y^{2}$$

Step-by-Step Solution

Answer

The expression factors to \((3x - 5y)(3x + 5y)\).

Step 1: Identify the Formula

The difference of squares formula is essential here. It states that \( a^2 - b^2 = (a - b)(a + b) \). The exercise asks us to factor the expression using this pattern.

Step 2: Express Terms as Squares

We need to express each term in the given expression as a square. The expression is \( 9x^2 - 25y^2 \). Observe that \( 9x^2 \) can be written as \( (3x)^2 \) and \( 25y^2 \) as \( (5y)^2 \).

Step 3: Apply the Formula

Now apply the difference of squares formula. Replace \( a \) with \( 3x \) and \( b \) with \( 5y \) in the formula \( a^2 - b^2 = (a - b)(a + b) \).

Step 4: Write the Solution

Substitute \( 3x \) and \( 5y \) in the formula, giving us \( (3x - 5y)(3x + 5y) \) as the factorized form of \( 9x^2 - 25y^2 \).

Key Concepts

Factoring

Factoring is a fundamental skill in algebra that involves breaking down complex expressions into simpler components or factors. Think of it like breaking apart a big puzzle into smaller, more manageable pieces that, when multiplied together, will form the original expression again.
Understanding how to factor polynomials efficiently can make solving algebraic equations much easier. There are different techniques used to factor various types of algebraic expressions, one of which is the difference of squares used in our exercise.
In the exercise, we focused on the expression \(9x^2 - 25y^2\). This is recognized as a difference of squares because both terms \(9x^2\) and \(25y^2\) can be represented as squares of \(3x\) and \(5y\) respectively. This type of factoring is straightforward when we know the pattern: \(a^2 - b^2 = (a - b)(a + b)\). By identifying these squares, we can swiftly derive the factors, in this case, \((3x - 5y)(3x + 5y)\).
Factoring expressions using the difference of squares method is not only practical but can help simplify many algebra problems, providing a more straightforward route to solutions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition, subtraction, multiplication, and division) that together represent a value. These expressions can take simple forms, like \(2x + 3\), or more complex forms, involving multiple terms and powers, such as \(9x^2 - 25y^2\).
Understanding algebraic expressions is crucial because they form the basis of algebra and are used to represent real-world problems and scenarios. Moreover, manipulating these expressions by applying operations and factoring techniques lets us solve equations and inequalities.
In our exercise, \(9x^2 - 25y^2\) is an algebraic expression utilizing subtraction between two squared terms, which allows us to apply the difference of squares pattern. Recognizing how each part of an expression relates to mathematical operations can make breaking down tasks into smaller steps much easier.

Polynomials

Polynomials are a specific type of algebraic expression consisting of terms that include variables raised to whole number powers and coefficients. A polynomial can have one or more terms, such as a monomial like \(5x^3\), a binomial like \(9x^2 - 25y^2\), or even a trinomial. Reacting strongly to negative powers and variables in the denominator, true polynomials exhibit a distinct structural simplicity.
In algebra, polynomials are operated upon using addition, subtraction, multiplication, and division. Factoring polynomials, like seen in the exercise, is an essential skill as it transforms polynomials into products of simpler polynomials or factors.
For example, in the polynomial \(9x^2 - 25y^2\), recognizing it as a binomial with a special pattern (difference of squares) allows us to factor it efficiently into \((3x - 5y)(3x + 5y)\). This highlights the importance of identifying specific polynomial forms and utilizing appropriate factoring techniques to simplify them.