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 polynomial for each expression in Column I. Notice that these are quadratic expressions in terms of x and y, and some resemble perfect square trinomials or difference of squares.
2. Recall the formulas for factoring: - Perfect square trinomial: \(a^2 + 2ab + b^2 = (a + b)^2\) - Perfect square trinomial with subtraction: \(a^2 - 2ab + b^2 = (a - b)^2\) - Difference of squares: \(a^2 - b^2 = (a + b)(a - b)\)
3. For expression a: \(x^2 + 10xy + 25y^2\), recognize it as a perfect square trinomial where \(a = x\) and \(b = 5y\). Use the formula \(a^2 + 2ab + b^2 = (a + b)^2\) to factor it.
4. For expression b: \(x^2 - 10xy + 25y^2\), recognize it as a perfect square trinomial with subtraction, where \(a = x\) and \(b = 5y\). Use the formula \(a^2 - 2ab + b^2 = (a - b)^2\) to factor it.
5. For expressions c and d, identify them as difference of squares: - For c: \(x^2 - 25y^2\), write it as \(a^2 - b^2\) with \(a = x\) and \(b = 5y\). - For d: \$25y^2 - x^2\(, write it as \)a^2 - b^2\( with \)a = 5y\( and \)b = x\(. Then factor both using \)a^2 - b^2 = (a + b)(a - b)$.

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 essential for factoring expressions like x² - 25y², which can be factored into (x + 5y)(x - 5y). It applies when two perfect squares are subtracted.

Matching Polynomials to Factored Forms

Matching involves identifying the correct factored form of a polynomial by recognizing patterns such as perfect squares or difference of squares. This skill helps in pairing expressions from one column to their equivalent factored forms in another.