Problem 127 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

A perfect square trinomial is a trinomial of the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\) and it is factored as \((a+b)^2\) or \((a-b)^2\), respectively.

Step 1: Define Perfect Square Trinomial

A perfect square trinomial is a type of trinomial which can be factored into a binomial squared. It has the form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\). The first and third terms are square numbers, and the middle term is twice the product of square roots of the first and third terms.

Step 2: Identify Perfect Square Trinomial

To identify a perfect square trinomial, check if the first and third terms are square numbers and if the middle term is twice the product of their square roots. If it matches either form \(a^2 + 2ab + b^2\) or \(a^2 - 2ab + b^2\), then it is a perfect square trinomial.

Step 3: Factor the Perfect Square Trinomial

To factor a perfect square trinomial, find the square roots of the first and third terms. The binomial is then the square root of the first term followed by the sign of the middle term and then the square root of the third term. The perfect square trinomial \(a^2 + 2ab + b^2\) is factored as \((a+b)^2\), and \(a^2 - 2ab + b^2\) is factored as \((a-b)^2\). Both factored forms are squared because the trinomial is a perfect square.