Problem 105 - College Algebra

Question

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

Step-by-Step Solution

Answer

A perfect square trinomial takes the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). It is factored into \((a + b)^2\) or \((a - b)^2\) by identifying and verifying the patterns and relationships among the terms within the trinomial.

Step 1: Understanding a Perfect Square Trinomial

A perfect square trinomial is the square of binomials. It takes the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\), where the first term is the square of the first element in the binomial, the second term is twice the product of the elements in the binomial and third term is the square of the second element in the binomial.

Step 2: Factoring a Perfect Square Trinomial

When factoring a perfect square trinomial, one must look for the pattern in the trinomial. If the expression being considered has the right format, it will factor into \((a + b)^2\) or \((a - b)^2\), where 'a' and 'b' are the squareroots of the first and third term respectively.

Step 3: Example of Factoring a Perfect Square Trinomial

Consider the trinomial \(x^2 + 6x + 9\). \(x^2\) is the square of \(x\), \(9\) is the square of \(3\), and the middle term \(6x\) is twice the product of \(x\) and \(3\). Therefore, recognizing the pattern, this trinomial factors into \((x + 3)^2\)