Would you show me how to completely factor each of the following expressions.
t^2 + 4tv + 4v^2
z^2 + 11z -42
Thank you for your help.
(t+2v)(t+2v)
(z-3)(z+14)
Certainly! To completely factor each of the given expressions, we will use different factoring techniques depending on their format.
Let's start with the first expression: t^2 + 4tv + 4v^2.
This expression is a perfect square trinomial because it can be written as (t + 2v)^2. Here's how to determine this:
1. Look at the first term, t^2. Find its square root, which is t.
2. Look at the last term, 4v^2. Find its square root, which is 2v.
3. Write down the square root of the first term and the square root of the last term: (t + 2v).
4. Square the expression written in the previous step: (t + 2v)^2.
So, the completely factored form of the expression t^2 + 4tv + 4v^2 is (t + 2v)^2.
Now let's move on to the second expression: z^2 + 11z - 42.
To factor this trinomial, you need to find two numbers whose sum is 11 (the coefficient of z in the middle term) and whose product is -42 (the constant term). The two numbers are 14 and -3 because 14 + (-3) = 11, and 14 * (-3) = -42.
To factor this expression, follow these steps:
1. Split the middle term (11z) into two terms using the two numbers you found: z^2 + 14z - 3z - 42.
2. Group the four terms and factor the common terms for each pair: z(z + 14) - 3(z + 14).
3. Notice that in both pairs, there is a common term, (z + 14), which can be factored out.
4. Write down the factored form: (z - 3)(z + 14).
Therefore, the completely factored form of the expression z^2 + 11z - 42 is (z - 3)(z + 14).
I hope this explanation helps you understand the factoring process! If you have any further questions, feel free to ask.