Prime factors of an expression

[Jehannum]

I'm reading a book that mentions writing an algebraic expression in terms of its prime factors, for example:

x² - 2 x - 3 = (x + 1) (x - 3)​

I know what 'prime factors' means for a number but not for an expression. Aren't these just 'factors'?

 

[PeroK]

I'm reading a book that mentions writing an algebraic expression in terms of its prime factors, for example:

x² - 2 x - 3 = (x + 1) (x - 3)​

I know what 'prime factors' means for a number but not for an expression. Aren't these just 'factors'?

They are factors, but they are also irreducible, as they themselves cannot be factored any further.

 

[Jehannum]

Ah ... thank you.

 

[fresh_42]

I'm reading a book that mentions writing an algebraic expression in terms of its prime factors, for example:

x² - 2 x - 3 = (x + 1) (x - 3)​

I know what 'prime factors' means for a number but not for an expression. Aren't these just 'factors'?

No, factors alone would be insufficient to describe what is meant. E.g. take $x^2+1=(x-i)(x+i)$. What are the factors here and which one is prime? The example shows that it depends on the ring you consider, i.e. the place in which the objects, here polynomials live.
In general a prime is defined by $p \textrm{ isn't a unit (invertible) and } (\,p \,\vert \,ab \Longrightarrow p\,\vert \,a \textrm{ or } p\,\vert \,b\,)$. This is the definition for numbers and polynomials. (Of course you could also define them as those elements $p$ of a commutative ring $R$ with unity, for which $R/pR$ is an integral domain.) The point is in any case, that it is important where the elements are taken from. An easy example would be ordinary primes: $5$ is a prime in the integers, but it isn't as a real number.