Simplify the expression:
12t^2 + 12t + 3
over
4t^2 - 1
i know that 3 is the GCF in the numerator, but i am so lost!
Pull out the three in the numerator. Now, break the numberator into two factors, and factor the denominator (difference of two squares). I suspect you will find a common factor that divides out.
To simplify the expression (12t^2 + 12t + 3) / (4t^2 - 1), you are correct in identifying that 3 is the greatest common factor (GCF) in the numerator. However, to simplify the expression further, we need to factorize the numerator and the denominator completely.
Let's start by factoring the numerator: 12t^2 + 12t + 3.
Step 1: Take out the GCF, which is 3: 3(4t^2 + 4t + 1).
Now, let's factorize the denominator: 4t^2 - 1. This is a difference of squares.
Step 2: Apply the difference of squares formula: (a^2 - b^2) = (a + b)(a - b). In this case, let a = 2t and b = 1.
4t^2 - 1 becomes: (2t + 1)(2t - 1).
Now we have factored both the numerator and the denominator, giving us 3(4t^2 + 4t + 1) / (2t + 1)(2t - 1).
Since there is no further simplification possible, this is the simplified expression.