Work each problem. Match each polynomial in Column I with its factored form in Column II.

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Identify the type of each polynomial in Column I by recognizing patterns such as perfect square trinomials or difference of squares.

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 \(b = 5y\).

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 \(b = 5y\).

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 \(b = 5y\).

For polynomial d: \(25y^2 - x^2\), also a difference of squares but with reversed terms, factor it similarly as \(a^2 - b^2 = (a+b)(a-b)\) where \(a = 5y\) and \(b = x\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 = x^2 + 2ax + a^2. Recognizing these allows you to factor expressions like x^2 + 10xy + 25y^2 into (x + 5y)^2 efficiently.

Difference of Squares

The difference of squares formula states that a^2 - b^2 = (a + b)(a - b). This concept helps factor expressions like x^2 - 25y^2 into (x + 5y)(x - 5y), by identifying the two terms as perfect squares separated by subtraction.

Matching Polynomials to Factored Forms

Matching involves recognizing the structure of polynomials and associating them with their equivalent factored expressions. This requires understanding factoring patterns and comparing terms to select the correct binomial factors from given options.