Work each problem. Match each polynomial in Column I with its factored form in Column II. a. x^2 + 10xy +25y^2 b. x^2-10xy +25y^2 c.x^2-25y^2 d. 25y^2-x^2 A. (x+5y ) (x-5y) B. (x+5y)^2 C. (x-5y)^2 D. (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. For example, the polynomial x^2 + 10xy + 25y^2 can be factored as (x + 5y)^2.

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 form is a^2 ± 2ab + b^2, which factors to (a ± b)^2. Recognizing these patterns is crucial for quickly factoring expressions like x^2 + 10xy + 25y^2, which fits the perfect square trinomial pattern and factors to (x + 5y)^2.

Difference of Squares

The difference of squares is a factoring technique used for expressions in the form a^2 - b^2, which factors to (a + b)(a - b). This concept is vital for polynomials like x^2 - 25y^2, where it can be factored into (x + 5y)(x - 5y). Understanding this concept allows for efficient simplification and solving of polynomial equations.