# Proof of prime factorization of an algebraic expression.

• jcoughlin
In summary: So, in summary, by using the Factor Theorem and demonstrating that (x-1) is a factor of (x^n-1), we can show that 11 is a factor of 12^n-1, which is the same as showing that the prime factorization of 22n*3n-1 includes 11 as one of its prime factors.
jcoughlin

## Homework Statement

Claim: If n is a positive integer, the prime factorization of 22n * 3n - 1 includes 11 as one of the prime factors.

## Homework Equations

Factor Theorem: a polynomial f(x) has a factor (x-k) iff f(k)=0.

## The Attempt at a Solution

First, we show that (x-1) is a factor of (xn-1). Let f(x)=xn-1, and k=1; then f(k)=0, and thus by the factor theorem (x-1) is a factor of (xn-1).

Next, consider 22n * 3n - 1 rewritten as 12n-1. As previously demonstrated, x-1 is a factor of xn-1. Letting x=12, we see that (12-1)=11 is a factor of 12n-1 for n>0.

Is this sufficient? Or do I need to go further than proving 11|(12n-1) to show that the prime factorization of 22n * 3n - 1 includes 11 as one of the prime factors?

Thanks,
James

Last edited:
I would add the statement that your number is 12^n - 1 somewhere.
If your number has 11 as a factor, this is one of the prime factors - it should be obvious that 11 is a prime, but you can write it down as well.

mfb said:
I would add the statement that your number is 12^n - 1 somewhere.
If your number has 11 as a factor, this is one of the prime factors - it should be obvious that 11 is a prime, but you can write it down as well.

Ah sorry, had a typo in the line that established 12^n-1 = 2^2n * 3^n - 1 .

Is showing 11 to be a factor, which happens to be prime, the same thing as showing that the prime factorization includes 11?

As prime factorization is unique, yes.

## 1. What is the proof of prime factorization of an algebraic expression?

The proof of prime factorization of an algebraic expression is a mathematical process that shows how to break down the expression into its prime factors. This means finding the unique combination of prime numbers that, when multiplied together, give the original expression.

## 2. Why is it important to prove the prime factorization of an algebraic expression?

Proving the prime factorization of an algebraic expression is important because it allows us to simplify complex expressions and understand their underlying structure. It also helps us to solve equations and perform other operations on the expression more easily.

## 3. How do you prove the prime factorization of an algebraic expression?

To prove the prime factorization of an algebraic expression, you can use various methods such as the factor tree method, the division method, or the prime factorization algorithm. The method you choose will depend on the complexity of the expression and your personal preference.

## 4. Can you give an example of proving the prime factorization of an algebraic expression?

Sure, let's take the expression 12x2y3. First, we can use the factor tree method to break it down into its prime factors: 12 = 2 x 2 x 3, x2 = x x x, and y3 = y x y x y. Then, we can combine the common factors and write the prime factorization as 22 x 3 x x2 x y3.

## 5. What are some real-life applications of proving the prime factorization of an algebraic expression?

Proving the prime factorization of an algebraic expression is used in various fields such as cryptography, number theory, and computer science. It also has practical applications in simplifying equations in physics and engineering, as well as in simplifying financial calculations and analyzing data in business and economics.

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