1:  The formula for the difference of two squares is a2 – b2 = (a+b)(a-b). To factor 81x2 – 1 you first write it as the difference of two squares. In the expression 81x2 -1, identify a and b; a=9x and b=1. (9x)2-12 is in the form of a2-b2, so replace a with 9x and replace b with 1 in the formula for the difference of two squares.

A2 – b2 = (a+b)(a-b)

(9x)2 – 12 = (9x+1)(9x-1)

2:

Squaring a binomial creates a perfect square trinomial: (a + b)2 (a – b)2 (a + b)2 = a2 + 2ab + b2 (a – b)2 = a2 – 2ab + b2

Factor: x2 + 12x + 36 Solution: Does this fit the pattern of a perfect square trinomial?

Yes. Both x2 and 36 are perfect squares, and 12x is twice the product of x and 6.

Since all signs are positive, the pattern is (a + b)2 = a2 + 2ab + b2. Let a = x and b = 6.

Answer: (x + 6)2 or (x + 6)(x + 6)

3:36a^2-60a+25. When factoring a trinomial, the first step is to factor out any common factors. The trinomial, 36a^2-60a+25, doesn’t have any common factors besides 1. The next step is to determine if the trinomial is a perfect square trinomial. A trinomial is a perfect square when two terms, a^2 and b^2, are squares and the other term is 2*a*b or -2*a*b, this term is twice the product of a and b. 36a^2=(6a)^2 and 25=(5)^2 are perfect squares and -60a=-2*6a*5,36a^2-60a+25, is a perfect square trinomial. A perfet square trinomial of the form a^2-2ab+b^2 is factored as (a-b)^2.  36a^2-60a+25=(6a)^2*5+5^2= (6a-5)^2