CONCEPT PREVIEW Work each problem.Match each polynomial in Column I with its factored form in Column II. I IIa. x² + 10xy + 25y² A. (x + 5y) (x - 5y)b. x² - 10xy + 25y² B. (x + 5y)²c. x² - 25y² C. (x - 5y)²d. 25y² - x² D. (5y + x) (5y - x)

Verified step by step guidance

1. Identify the structure of each polynomial in Column I to determine if they are perfect square trinomials or differences of squares.
2. For polynomial a: \(x^2 + 10xy + 25y^2\), recognize it as a perfect square trinomial, which can be factored as \((x + 5y)^2\).
3. For polynomial b: \(x^2 - 10xy + 25y^2\), recognize it as a perfect square trinomial, which can be factored as \((x - 5y)^2\).
4. For polynomial c: \(x^2 - 25y^2\), recognize it as a difference of squares, which can be factored as \((x + 5y)(x - 5y)\).
5. For polynomial d: \(25y^2 - x^2\), recognize it as a difference of squares, which can be factored as \((5y + x)(5y - x)\).

Key Concepts

Factoring Polynomials

Factoring polynomials involves rewriting a polynomial as a product of its simpler components, or factors. This process is essential for solving polynomial equations and simplifying expressions. Common techniques include identifying perfect squares, using the difference of squares, and applying the distributive property. Understanding how to factor polynomials allows for easier manipulation and analysis of algebraic expressions.

Perfect Square Trinomials

A perfect square trinomial is a specific type of polynomial that can be expressed as the square of a binomial. The general forms are (a + b)² = a² + 2ab + b² and (a - b)² = a² - 2ab + b². Recognizing these patterns is crucial for quickly factoring expressions like x² + 10xy + 25y², which can be factored as (x + 5y)². This concept simplifies the factoring process significantly.

Difference of Squares

The difference of squares is a factoring technique used for expressions in the form a² - b², which can be factored into (a + b)(a - b). This concept is particularly useful for polynomials like x² - 25y² and 25y² - x², as they can be rewritten using this identity. Understanding the difference of squares is essential for efficiently solving polynomial equations and recognizing factorable forms.