I do not understand special factors, the teacher on Friday tried to explain in to me in class but she went so fast, and every time we complain that we don't understand it she gets mad. So could someone please help me with at least one problem so I can understand it better, and please explain it step by step.

Directions: Determine whether each trinomial is a perfect square trinomial. If so factor it.

1. x^2-16x+64

I think its a perfect square trionomial because you can square 4^2 to get 16 and 8^2 to get 64

(x - 8)(x - 8 )

first condition:

the first and last terms must be perfect squares, and they must both be positive
second condition:
the square root of the first term times the square root of the last term, then doubled, should be equal to the middle term.

e.g.
25x^2 + 60xy + 36y^2

square root of first: 5x
square root of last : 6y
product: 30xy
doubled: 60xy , so YES

e.g.
x^4 - 10x^2 + 25

square root of first: x^2
square root of last: 5
product : 5x^2
doubled : 10x^2, YES!

does that make sense?

To determine if a trinomial is a perfect square trinomial, we need to check if it can be factored into the square of a binomial.

Let's take a look at the trinomial: x^2 - 16x + 64

Step 1: Check if the first term is a perfect square. In this case, the first term is x^2, which is a perfect square (x^2 = x*x).

Step 2: Check if the last term is a perfect square. In this case, the last term is 64, which is a perfect square (64 = 8*8).

Step 3: If both the first and last terms are perfect squares, then check if the middle term is twice the product of the square roots of the first and last terms. In this case, the middle term is -16x.

The square root of the first term, x^2, is x.
The square root of the last term, 64, is 8.

Twice the product of x and 8 is 2 * (x * 8) = 2 * 8x = 16x.

Since the middle term (-16x) is indeed twice the product of the square roots of the first and last terms (x and 8), the trinomial x^2 - 16x + 64 is a perfect square trinomial.

To factor it, we can rewrite it as the square of a binomial:

x^2 - 16x + 64 = (x - 8)^2

So, the factored form of the trinomial is (x - 8)^2.