Problem 127 - College Algebra

Question

What is a perfect square trinomial and how is it factored?

Step-by-Step Solution

Answer

A perfect square trinomial is a trinomial that can be factored into the square of a binomial. It has the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). When factoring, the trinomial \(x^2 + 2xy + y^2\) becomes \((x + y)^2\) and the trinomial \(x^2 - 2xy + y^2\) becomes \((x - y)^2\).

Step 1: Define Perfect Square Trinomial

A Perfect Square Trinomial is a trinomial that can be factored into the square of a binomial. It takes the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). In this form, \(a\) and \(b\) are real numbers and \(a\), being the coefficient of the squared first term, is a positive real number.

Step 2: Identify Perfect Square Trinomial

To identify if a trinomial is a perfect square, it must fit one of the two forms mentioned. If the trinomial is \(x^2 + 2xy + y^2\) or \(x^2 - 2xy + y^2\), then \(a = x\) and \(b = y\) or \(b = -y\). Check if the middle term is twice the product of the square roots of the first and third term.

Step 3: Factor Perfect Square Trinomial

After successfully identifying a perfect square trinomial, you can factor it into the square of a binomial. If the trinomial is in the form \(x^2 + 2xy + y^2\), it is factored as \((x + y)^2\), and if it is in the form \(x^2 - 2xy + y^2\), it is factored as \((x - y)^2\).