Question about dividing polynomials: f(x)=(x^47-1) and g(x)=(x^3-1)

[Pranjatheman]

Hi
So I need to divide these two polynomials f(x)=(x^47-1) and g(x)=(x^3-1)
The basic division takes too long ( I did get the end result), but is there a faster way for polynomials with bigger exponents?

 

[Otis]

I need to divide these two polynomials f(x)=(x^47-1) and g(x)=(x^3-1)

The basic division takes too long ( I did get the end result), but is there a faster way for polynomials with bigger exponents?

I'm guessing that "dividing these two" means f/g.

Did you factor g first?

I'm not sure how you carried out the division, but there's a method known as Synthetic Division, for dividing a polynomial by a linear factor.

 

[stapel]

So I need to divide these two polynomials f(x)=(x^47-1) and g(x)=(x^3-1)
The basic division takes too long ( I did get the end result), but is there a faster way for polynomials with bigger exponents?

Have you yet encountered the following result?

. . . . .\(\displaystyle \dfrac{x^n\, -\, 1}{x\, -\, 1}\, =\, x^{n-1}\, +\, x^{n-2}\,+\, ...\, +\, x^2\, +\, x\, +\, 1\)

This allows you to take the factorization of the divisor (which is a difference of cubes) and cancel out a common factor. Then do the remaining long division. I don't know that this will be particularly "shorter", though...

Note: If you run this through Wolfram's Alpha solver engine, the result is just plain messy. There may not be a "short" way of doing this. :shock: