Question

In writing a polynomial as the product of polynomials of lesser degrees, what does it mean to say that a factor is prime?

Short answer

In the context of polynomials, a factor is considered prime if it cannot be factored further, i.e., it can't be expressed as a product of two non-constant polynomials.

Step-by-step solution

Understanding prime numbers

In the world of integers, a prime number is any number that only has two distinct positive divisors, 1 and the number itself. For example, the first few primes are 2, 3, 5, 7, and 11.

Applying concept to polynomials

For a polynomial, a factor is considered prime if it cannot be factored further within the set of polynomials over which we are considering factorization. Basically it can't be represented as a product of two non-constant polynomials.

Practical example

For instance, consider the polynomial \( x^2 - 4 \). It can be factored into (x - 2)(x + 2). Here, the polynomials (x - 2) and (x + 2) are prime, because they cannot be factored further into polynomials of lesser degrees.