Polynomial division is a mathematical process used to divide one polynomial by another polynomial. It is similar to long division, but instead of dividing numbers, we are dividing terms in a polynomial expression.
In polynomial division, factors refer to the terms that can be multiplied together to obtain the original polynomial. In other words, they are the numbers or expressions that divide evenly into the polynomial without any remainder.
To find the factors of a polynomial in Q[x] (rational coefficients), you can use the rational root theorem. This theorem states that if a polynomial has rational roots, they will be in the form of p/q, where p is a factor of the constant term and q is a factor of the leading coefficient. By testing these possible roots, we can find the factors of the polynomial.
In Z5[x] (polynomials with coefficients in the ring of integers mod 5), we can use the same method as in Q[x], but we only need to test the possible roots 0, 1, 2, 3, and 4 since these are the only integers in Z5. Additionally, we can also use the factor theorem, which states that if a polynomial has a root r, then (x-r) is a factor of the polynomial.
To divide a polynomial, we use the factors we have found to perform long division. We start by dividing the highest degree term of the polynomial by the highest degree factor, and then continue with the remainder of the polynomial. This process is repeated until there is no remainder, and the quotient is the result of the polynomial division.