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Factorisation is the process of breaking down a polynomial into smaller polynomials, known as factors, that when multiplied together, produce the original polynomial.
Factorisation allows us to simplify complicated polynomials and solve equations more easily. It also helps us identify the roots of a polynomial, which are the values that make the polynomial equal to zero.
The main difference between factorisation in Q[x] and Z[x] is the coefficients used. In Q[x], the coefficients can be any rational number, while in Z[x], the coefficients are restricted to integers.
To factorise a polynomial in Q[x] and Z[x], we first look for any common factors among the terms. Then, we use various techniques such as grouping, difference of squares, and trial and error to factor the polynomial into smaller polynomials.
No, not all polynomials in Q[x] and Z[x] can be factorised. Some polynomials may not have any factors or may only have complex factors. In these cases, the polynomial is considered to be irreducible and cannot be broken down into simpler polynomials.
