Problem 127 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

A perfect square trinomial is a polynomial that can be written as \(a^2+2ab+b^2\) or \(a^2-2ab+b^2\). It is called a perfect square trinomial because its first and third terms are perfect squares and the middle term is twice the product of the square roots of the first and third term. To factor it, recognise if the trinomial fits the perfect square criteria and if the middle sign is positive then it can be factored to \((a+b)^2\), or if negative then it can be factored to \((a-b)^2\).

Step 1: Defining the Term

A perfect square trinomial is a type of polynomial that can be written in the form \(a^2+2ab+b^2\) or \(a^2-2ab+b^2\). It can also be represented as \((a+b)^2\) or \((a-b)^2\). It is called 'perfect square' because it is the square of a binomial.

Step 2: Characteristics of a Perfect Square Trinomial

There are two main characteristics of a perfect square trinomial. First, the first and third terms are perfect squares. The second characteristic is that the middle term is twice the product of the square roots of the first and third terms.

Step 3: Factoring a Perfect Square Trinomial

The process of factoring a perfect square trinomial involves the following steps: 1. Identify if the trinomial is a perfect square. This can be done by confirming that the first and third terms are perfect squares and the middle term is twice the product of the square roots of first and third term. 2. If the trinomial fits the criteria, then the trinomial can be factored into \((a+b)^2\) if the sign in the center of the trinomial is positive, and it can be factored into \((a-b)^2\) if the sign in the center of the trinomial is negative.