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

Verified step by step guidance

1. Identify the type of each polynomial in Column I by recognizing common factoring patterns such as perfect square trinomials and difference of squares.
2. For polynomial a: \(x^{2} + 10xy + 25y^{2}\), check if it fits the form of a perfect square trinomial \(a^{2} + 2ab + b^{2} = (a + b)^{2}\). Here, \(a = x\) and \(b = 5y\).
3. For polynomial b: \(x^{2} - 10xy + 25y^{2}\), check if it fits the form of a perfect square trinomial \(a^{2} - 2ab + b^{2} = (a - b)^{2}\). Again, \(a = x\) and \(b = 5y\).
4. For polynomial c: \(x^{2} - 25y^{2}\), recognize it as a difference of squares, which factors as \(a^{2} - b^{2} = (a + b)(a - b)\) with \(a = x\) and \(b = 5y\).
5. For polynomial d: \$25y^{2} - x^{2}\(, also a difference of squares but with terms reversed, factor it similarly as \)(5y + x)(5y - x)$.

Key Concepts

Factoring Perfect Square Trinomials

A perfect square trinomial is a quadratic expression that can be written as the square of a binomial, such as (x + a)² = x² + 2ax + a². Recognizing these allows quick factoring of expressions like x² + 10xy + 25y² into (x + 5y)².

Difference of Squares

The difference of squares formula states that a² - b² = (a + b)(a - b). This is used to factor expressions like x² - 25y² into (x + 5y)(x - 5y), by identifying the two terms as perfect squares.

Matching Polynomials to Factored Forms

Matching involves comparing given polynomials with their factored equivalents by recognizing patterns such as perfect squares or differences of squares. This skill helps in quickly identifying the correct factorization from multiple choices.