What is the Greatest Common Divisor of Two Polynomials?

In summary, the greatest common divisor of X^a - 1 and X^b - 1, where (a,b) belong to the set of natural numbers, can be found using Euclid's algorithm on a and b. The last non-zero remainder of the algorithm, which is equivalent to the greatest common divisor of a and b, will also be the greatest common divisor of X^a - 1 and X^b - 1. It is recommended to use \gcd instead of \text{gcd} in MathJax and LaTeX for proper spacing in expressions.
  • #1
geoffrey159
535
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Homework Statement


What is the greatest common divisor of ##X^a - 1 ## and ## X^b - 1 ##, ##(a,b) \in \mathbb{N}^\star## ?

Homework Equations

The Attempt at a Solution



Assuming that ## a\le b ##, I find by euclidian division of ##b## by ##a## that

## b = an + r \Rightarrow X^b - 1 = (X^a-1) (\sum_{k=1}^{n} X ^ {b-ka}) + X ^r - 1 ##

So ##\text{gcd}(X^b - 1,X^a - 1) = \text{gcd}(X^a - 1,X^r - 1) ##
So if I apply Euclid's algorithm on ##a## and ##b##, I will automatically get that the last non-zero remainder, which is ## \text{gcd}(a,b)##, guarantees that ## X ^{ \text{gcd}(a,b) } - 1 ## is the last non-zero remainder of the algorithm applied on ##X^a - 1 ## and ## X^b - 1 ##, and therefore is their greatest common divisor. Right ?
 
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  • #2
Yes, It is the greatest commun divisor :)
 
  • #3
OK thank you ;-)
 
  • #4
geoffrey159 said:
So ##\text{gcd}(X^b - 1,X^a - 1) = \text{gcd}(X^a - 1,X^r - 1) ##

In MathJax and LaTeX, don't write \text{gcd}; instead write \gcd. Unlike \text{gcd} this will yield proper spacing in expressions like ##8\gcd(a,b)##, whereas with \text{gcd} you'll see ##8\text{gcd}(a,b)## instead, with the conspicuous lack of spacing. And the amount of space to the left and right of ##\gcd## will actually depend on the context.
 

FAQ: What is the Greatest Common Divisor of Two Polynomials?

1. What is the definition of arithmetic on polynomials?

Arithmetic on polynomials involves performing mathematical operations such as addition, subtraction, multiplication, and division on expressions that contain variables and coefficients. These expressions are called polynomials.

2. How do you add or subtract polynomials?

To add or subtract polynomials, you must first arrange the terms in descending order of their exponents. Then, simply combine like terms by adding or subtracting the coefficients of the terms with the same variable and exponent.

3. What is the FOIL method for multiplying polynomials?

The FOIL method is a mnemonic device used to remember the steps for multiplying two binomials. It stands for First, Outer, Inner, Last. This means that you multiply the first terms, then the outer terms, then the inner terms, and finally the last terms. Then, combine like terms to simplify the expression.

4. Can polynomials be divided?

Yes, polynomials can be divided, but the process can be more complex than addition, subtraction, and multiplication. To divide polynomials, you must use long division or synthetic division. These methods involve dividing the terms of the polynomials, similar to how you would divide numbers in long division.

5. What is the degree of a polynomial?

The degree of a polynomial is the highest exponent in the polynomial expression. For example, the polynomial 3x^2 + 5x + 1 has a degree of 2 because the term with the highest exponent is x^2. The degree is important because it determines the behavior of the polynomial, such as its end behavior and the number of roots it has.

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