Factorization of 36^x-9^x-4^x+1
In this article, we will explore the factorization of the expression 36^x-9^x-4^x+1. This expression may seem complex at first, but it can be simplified using some clever algebraic manipulations.
Step 1: Factor out a common term
Let's start by factoring out the common term 9^x from the first two terms:
36^x - 9^x - 4^x + 1 = (9^x)(4^x - 1) - 4^x + 1
Step 2: Factor the quadratic expression
Next, we can factor the quadratic expression inside the parentheses:
4^x - 1 = (2^x + 1)(2^x - 1)
So, our expression becomes:
(9^x)((2^x + 1)(2^x - 1)) - 4^x + 1
Step 3: Simplify the expression
Now, let's simplify the expression by combining like terms:
(9^x)(2^x + 1)(2^x - 1) - 4^x + 1 = (9^x)(2^2x - 1) - 4^x + 1
= (2^2x)^x(2^2x - 1) - 4^x + 1 (since 9 = 3^2)
= (4^x)^2(2^2x - 1) - 4^x + 1
Final Answer
After simplifying, we get the final factorization:
36^x - 9^x - 4^x + 1 = (4^x - 1)(4^x + 1)(2^2x - 1)
This may not be the simplest form, but it's a significant improvement from the original expression.
