Problem 18 - Elementary Algebra

Question

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

Step-by-Step Solution

Answer

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

Step 1: Identify the Expression Form

Recognize that the expression is a difference of squares, which can be expressed in the form \( a^2 - b^2 \). In the given problem, \( 25 \) is a perfect square \( (5^2) \) and \( 9y^2 \) is another perfect square \( (3y)^2 \). Therefore, the expression can be written as \((5)^2 - (3y)^2 \).

Step 2: Apply the Difference of Squares Formula

Recall the difference of squares formula: \( a^2 - b^2 = (a - b)(a + b) \). Substitute \( a = 5 \) and \( b = 3y \) from the previous step into this formula. Therefore, \( (5)^2 - (3y)^2 = (5 - 3y)(5 + 3y) \).

Step 3: Write the Final Factored Form

Combine the results from applying the difference of squares formula to express the original expression. Thus, \( 25 - 9y^2 \) factors completely to \( (5 - 3y)(5 + 3y) \).

Key Concepts

Factoring Algebraic Expressions

Factoring is a crucial skill when handling algebraic expressions. It involves breaking down complex expressions into simpler components or factors. This can make solving algebraic equations more manageable and provide insight into the properties of the expression itself.

An essential technique in factoring is recognizing special patterns, such as the difference of squares. This pattern appears when an expression can form as \( a^2 - b^2 \). The difference of squares is special because it can be easily factored into the expression \((a - b)(a + b)\).

To factor using this technique, identify the square components of your expression. For instance, in the expression \( 25 - 9y^2 \), both 25 and \( 9y^2 \) are perfect squares. As a result, you replace \( 25 \) with \( (5)^2 \) and \( 9y^2 \) with \( (3y)^2 \). Recognizing such forms can significantly simplify the process of factoring in algebra.

Algebraic Equations

Algebraic equations consist of mathematical statements that assert equality between two expressions. They often involve variables and constants and allow us to find unknown quantities.

Understanding how to manipulate algebraic equations is vital. This includes operations like addition, subtraction, and importantly, factoring. Factoring serves as an algebraic tool to simplify expressions and solve equations by breaking them into multiplicative components. For example, when solving an equation, factoring can reveal solutions by showing root values where the equation holds true.

In scenarios like the difference of squares, recognizing the pattern enables quick factorization, such as turning \( 25 - 9y^2 \) into \( (5 - 3y)(5 + 3y) \). This helps simplify and solve equations faster when you set the expression equal to another value, often zero.

Polynomials

Polynomials are algebraic expressions that consist of variables raised to whole number powers and their coefficients. They are foundational elements in algebra with various forms, from simple monomials to complex multinomials.

The expression \( 25 - 9y^2 \) is a polynomial of the second degree or a quadratic polynomial. Recognizing its structure helps in factoring. Quadratic polynomials often fall into specific factorization patterns like the difference of squares or trinomials.

Factoring polynomials aids in understanding their roots, which are the values for which the polynomial equals zero. Recognizing patterns such as perfect squares or the difference of squares is central to simplifying polynomials, solving algebraic equations, and applying them in broader mathematical problems. Factoring transforms polynomials for ease of evaluation and manipulation.