Problem 11 - Elementary and Intermediate Algebra

Question

Factor. \(9 y^{2}-64\)

Step-by-Step Solution

Answer

The factorization of \(9y^2 - 64\) is \((3y - 8)(3y + 8)\).

Step 1: Identify the Expression Type

Notice that the expression \(9y^2 - 64\) fits the pattern of a difference of squares. A difference of squares is any expression of the form \(a^2 - b^2\), which can be rewritten as \((a-b)(a+b)\).

Step 2: Express Each Term as a Square

Determine what squares will combine into \(9y^2 - 64\). Recognize that \(9y^2\) is \((3y)^2\) because \((3y) \times (3y) = 9y^2\). Similarly, recognize that \(64\) is \(8^2\) because \(8 \times 8 = 64\).

Step 3: Apply the Difference of Squares Formula

Apply the difference of squares formula, \((a-b)(a+b)\), where \(a = 3y\) and \(b = 8\). Substitute these into the formula: \( (3y - 8)(3y + 8) \).

Step 4: Confirm the Factorization

To ensure the factorization is correct, expand \((3y - 8)(3y + 8)\) using the distributive property: \((3y)(3y) + (3y)(8) - (8)(3y) - (8)(8) = 9y^2 - 64 \). The expanded expression matches the original, confirming the factorization is correct.

Key Concepts

Understanding the Difference of Squares

The difference of squares is a specific pattern that occurs when you have an expression of the form \(a^2 - b^2\). This pattern is called the "difference of squares" because it involves subtracting one perfect square from another.

• Perfect Squares: Numbers like 4, 9, 16, and algebraic expressions like \(y^2\) or \((3y)^2\) are considered perfect squares because they can be written as something squared.
• Recognizing the Pattern: Notice the operation between the two squares is subtraction, hence the term "difference" in difference of squares.

To factor an expression that fits this pattern, remember: it forms two binomials \((a-b)(a+b)\). For example, start with \(9y^2 - 64\), recognize it's a difference of squares because \(9y^2 = (3y)^2\) and \(64 = 8^2\). Applying the pattern, we can factor it into \((3y - 8)(3y + 8)\). This method is efficient for quickly breaking down expressions involving perfect squares.

Algebraic Expressions Simplified

Algebraic expressions are combinations of numbers, symbols, or variables associated with operations like addition, subtraction, multiplication, and division. They serve as powerful tools for representing real-world problems in a mathematical format.

• Components: Expressions consist of terms, which are single numbers or variables multiplied together.
• Coefficients: Numbers that are multiplied by variables in terms (e.g., in \(3y\), 3 is the coefficient).
• Constant terms: Plain numbers without variables (e.g., 8 in the expression \(3y + 8\)).

The goal in working with algebraic expressions is usually to simplify or rearrange them in some way. For example, through factorization, you aim to express an algebraic statement in its simplest form. Factoring techniques, such as those used to handle differences of squares, transform the algebraic terms into a more insightful, workable expression.

The Art of Polynomial Factorization

Polynomial factorization is like discovering the building blocks of algebra. It involves breaking down a polynomial into simpler components, or factors, that multiply together to give the original expression. Here’s why it’s vital:

• Simplification: Turning complicated expressions into simple ones.
• Solving Equations: Factoring is often a crucial step in finding the roots (or solutions) of equations.
• Applications in Real Life: From physics to finance, factorization helps model and solve real-world problems.

To factor a polynomial, look for common patterns or divide by common terms. For instance, in the example \(9y^2 - 64\), recognizing it as a "difference of squares" allows quick factorization into \((3y - 8)(3y + 8)\). Factoring streamlines processes, enhances mathematical manipulation, and underpins much of algebraic problem-solving.