91Ó°ÊÓ

A perfect square trinomial is a special form of quadratic expression that can be easily factored. It follows a specific pattern:

• In the form \((ax^2 + 2abx + b^2)\)
• This can be factored into a binomial squared as \((ax + b)^2\)

Perfect square trinomials are characterized by having the first term as a square, the last term as a square, and the middle term as twice the product of the square roots of the first and last terms. For example, in the exercise expression a perfect square trinomial appears twice: \(a x^{2} - 2 a x y + a y^{2}\) and \(-x^{2} + 2 x y - y^{2}\). When factored, these become \((a(x-y)^2)\) and \(-((x-y)^2)\), respectively. Recognizing and factoring perfect square trinomials makes simplifying complex expressions much easier.

Finding a common factor is a key technique in algebra. It helps take apart expressions to a more simplified form. In essence, it involves identifying a term or factor that appears in multiple parts of an expression.

In the given exercise, after factoring both segments into \((a(x-y)^2)\) and \(-((x-y)^2)\), it is noticeable that \((x-y)^2\) is a common factor. A common factor can be factored out using parentheses. This process not only simplifies the expression but clearly reveals any underlying structure. It's much like pulling out the common pieces in a puzzle and observing the pattern.Finding common factors is crucial, particularly in expressions where terms might initially seem unrelated.

The distributive property is a fundamental algebraic principle used in simplifying expressions. The property states that multiplying a single term by a sum of terms inside parenthesis can be expanded distributively:

• \(a(b + c) = ab + ac\)

In this exercise, once it is evident that \((x-y)^2\) is a common factor, the expression \(a(x-y)^2 - (x-y)^2\) becomes ripe for the distributive property.

By factoring out \((x-y)^2\), the distributive property lets us simplify the expression to \((a-1)(x-y)^2\). This isolates the common factor while reducing the expression to a simpler form. The distributive property is essential in algebraic manipulation, showcasing the elegance of mathematical transformations.