The Correct Answer and Explanation is:
The correct and complete factorization of the expression is (3q + 1)².
Explanation
The problem requires us to factor the quadratic trinomial 9q² + 6q + 1. Factoring is the process of breaking down a polynomial into a product of simpler expressions. In this case, we can identify the expression as a special type known as a perfect square trinomial.
A perfect square trinomial has a specific structure: a² + 2ab + b² or a² – 2ab + b². The first form factors into (a + b)², and the second factors into (a – b)². We can check if our expression, 9q² + 6q + 1, fits the first pattern.
First, let’s examine the first term, 9q². This term is a perfect square because the square root of 9 is 3 and the square root of q² is q. Therefore, it can be written as (3q)². This suggests that our ‘a’ term in the formula is 3q.
Next, we examine the last term, 1. This is also a perfect square, as the square root of 1 is 1. It can be written as 1². This suggests that our ‘b’ term in the formula is 1.
The final step is to verify if the middle term, 6q, matches the ‘2ab’ part of the pattern. We can test this by multiplying 2 by our identified ‘a’ (3q) and ‘b’ (1). The calculation is 2 * (3q) * 1 = 6q. This result matches the middle term of the original expression exactly.
Since all three parts of the expression fit the pattern a² + 2ab + b², we can conclude that 9q² + 6q + 1 is a perfect square trinomial. The factored form is (a + b)², which for our values becomes (3q + 1)². This is the complete factorization because the binomial (3q + 1) cannot be factored further.
