Problem 112 - Introductory Algebra for College Students

Question

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

Step-by-Step Solution

Answer

A perfect square trinomial is a polynomial of the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). It can be factored as \((a+b)^2\) or \((a-b)^2\), respectively. For example, the trinomial \(x^2 + 6x + 9\) can be factored as \((x+3)^2\).

Step 1: Identify the Perfect Square Trinomial

A perfect square trinomial is a polynomial of the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). This form should be able to be written as \((a+b)^2\) or \((a-b)^2\) respectively.

Step 2: Understanding Factoring a Perfect Square Trinomial

Factoring a perfect square trinomial involves expressing the trinomial in one of the forms mentioned previously. Essentially, you are looking for values of \(a\) and \(b\) such that the trinomial fits the pattern of either \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). The trinomial can be factored as \((a+b)^2\) or \((a-b)^2\), respectively.

Step 3: Example of Factoring a Perfect Square Trinomial

For example, let's factor \(x^2 + 6x + 9\). We can see that the trinomial follows the pattern \(a^2 + 2ab + b^2\). By comparing each term in the trinomial to the pattern, we can see that \(a=x\) and \(b=3\), therefore the trinomial can be factored as \((x+3)^2\).