How do I completely factor this expression?
256z^2 -4 -192z^2 +3
I appreciate your help.
64 z^2 - 1
That is the difference of two squares
remember
a^2-b^2 = (a-b)(a+b)
so
(8z-1)(8z+1)
256z^2 -4 -192z^2 +3
= (256z^2-4) - 3(64z^2-1)
= (16z-2)(16z+2)-3(8z-1)(8z+1)
= 4(8z-1)(8z+1)-3(8z-1)(8z+1)
= (8z-1)(8z+1)
cool, eh?
To completely factor the expression 256z^2 - 4 - 192z^2 + 3, you can follow these steps:
Step 1: Group the terms with common factors.
(256z^2 - 192z^2) + (-4 + 3)
Simplifying this, we have:
64z^2 - 1
Step 2: Try to identify if this expression can be factored further.
In this case, we have a difference of squares, since 64z^2 - 1 can be written as (8z)^2 - 1^2.
Step 3: Apply the formula for the difference of squares.
The difference of squares formula states that a^2 - b^2 can be factored as (a + b)(a - b).
In this case, a = 8z and b = 1, so we can factor 64z^2 - 1 as (8z + 1)(8z - 1).
Therefore, the completely factored expression is:
(8z + 1)(8z - 1).
Note: Factoring is not always possible or straightforward for every expression. In some cases, the expression may be prime or require more advanced techniques to factor completely.