Problem 20 - Introductory Algebra for College Students

Question

Factor each difference of two squares. $$9 x^{2}-25 y^{2}$$

Step-by-Step Solution

Answer

\(9x^2 - 25y^2\) factors to \((3x - 5y)(3x + 5y)\)

Step 1: Identify difference of squares

Identify \(9x^2 - 25y^2\) as a difference of squares. Here, \(a^2\) corresponds to \(9x^2\) and \(b^2\) corresponds to \(25y^2\). Therefore, \(a\) is \(3x\) and \(b\) is \(5y\).

Step 2: Apply difference of squares formula

Express \(9x^2 - 25y^2\) as \((3x)^2 - (5y)^2\). Then, apply the difference of squares formula \((a-b)(a+b)\).

Step 3: Factor the expression

Substitute \(3x\) for \(a\) and \(5y\) for \(b\) in the difference of squares formula to get \( (3x - 5y)(3x + 5y)\).

Key Concepts

Factoring Polynomials

Factoring polynomials is a fundamental skill in algebra which involves breaking down a polynomial into simpler multiplicative components. This makes evaluating, simplifying, and solving polynomial equations much easier. When dealing with a polynomial like the one in our exercise, such as a difference of squares, the goal is to express it in a factored form, which is a product of its binomial expressions.

Key steps in factoring any polynomial include:

• Identifying common factors that can be factored out from each term.
• Recognizing special polynomial forms, such as the difference of squares, perfect square trinomials, and sum or difference of cubes.
• Breaking down the polynomial using an appropriate factoring formula.

For the exercise given, recognizing the expression's pattern helped to apply the correct formula and simplify it as \( (3x - 5y)(3x + 5y) \). Successfully factoring polynomials using these strategies is crucial for problem-solving in algebra.

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and operators (like addition, subtraction, etc.). They are the building blocks of algebra \( - \) forming the basis for equations and inequalities. Understanding algebraic expressions involves knowing how each part contributes to the whole and how they interact within an equation or formula.

When working with algebraic expressions, it's essential to:

• Identify different terms and their coefficients.
• Recognize the power and role of variables within the expression.
• Apply operations across terms correctly according to algebraic rules.

In the case of our simple difference of squares \( 9x^2 - 25y^2 \), each part is clearly defined. The expression consists of two squared terms, each representing perfect squares of smaller expressions (\( 3x \) and \( 5y \)). By transforming it using algebraic rules, we unravel its structure into an understandable format. This understanding is crucial for further manipulation and problem-solving tasks.

Quadratic Expressions

Quadratic expressions are polynomial expressions where the highest power of the variable is two. These often take the form \( ax^2 + bx + c \), known as the standard quadratic form. However, they can also include special types such as the difference of squares, which is a modified quadratic expression.

To master quadratic expressions, one should:

• Identify the type of quadratic expression \( - \) whether it's in a standard form or a special form like the difference of squares.
• Apply appropriate factoring techniques, such as completing the square or using formulas like the difference of squares.
• Understand how these expressions model real-world problems and assist in finding their solutions.

Our expression \( 9x^2 - 25y^2 \) is distinguished as a difference of squares \( - \) a special case of quadratic expressions, allowing us to factor it into a product of binomials, \( (3x - 5y)(3x + 5y) \). Recognizing and correctly handling quadratic expressions is crucial in algebra and beyond, as they lay the groundwork for solving complex mathematical problems.