Final answer:
A perfect square trinomial can be factored into the square of a binomial by confirming the first and third terms are perfect squares and the middle term is double the product of the square roots of these terms. Recognizing perfect square trinomials is useful for simplifying the process of solving quadratic functions.
Step-by-step explanation:
Perfect Square Trinomials and Factoring
A perfect square trinomial is an expression that can be factored into the square of a binomial. To factor such a trinomial, you need to confirm that the first and third terms are perfect squares and that the middle term is twice the product of the square roots of these terms. For instance, if we have the trinomial a² + 2ab + b², we can factor it as (a + b)² because a² is a perfect square, b² is a perfect square, and 2ab is twice the product of a and b. This also applies to trinomials with a negative middle term, like a² - 2ab + b², which factors as (a - b)².
When working with polynomials, such as second-order polynomials or quadratic functions, recognizing perfect square trinomials can simplify the process of solving equations. For instance, the quadratic formula could be used, but factoring is often simpler when the equation forms a perfect square. Understanding how to factor perfect square trinomials is a fundamental skill in algebra that simplifies solving these types of quadratic equations.