A number can be factored into a product of its component factors
A number can be factored into a product of its prime .
But, What exactly is a prime number ?
Prime numbers are numbers greater than 1 that are evenly divisible only by themselves and 1

Is it a number that can only be evenly divided by that number itself and one ?As in like it cannot be factored further?

Yes, if one could factor primes further then they would be divisible by another number and hence wouldn't be primes. In fact beyond that every number can be uniquely factored into its components, which is one of the fundamental theorems of number theory.

The first few prime numbers are 2, 3, 5, 7, 11, 13 .
A prime number is a positive integer which has no factors other than 1 and itself. 1 itself, by definition, is not a prime number.
Prime numbers cant be divided any further and thus can be thought of as the atoms of numbers.
Any number which is not prime can be written as the product of prime numbers, we simply keep dividing it into more parts until all factors are prime

Yes and as I mentioned above every number can be factored into its prime factors in one way only. This is also the reason why 1 is not condsidered a prime number.

No, I do not think so. I believe that prime factorization is part of pre-algebra and number theory. The only connection I can think of is factorizing polynomials, which may involve thinking about the factors of a number.

I am not sure which type of questions you are looking for. If you are looking for questions, which involve simply factorizing numbers I would recommend a website called " Brilliliant.org". They offer problems and information on nearly all branches of maths and offer those kinds of practice problems.

Hope that helps. But apart from that, most computers and more modern calculators offer prime factorization programs. Google will factorize ridiculous numbers for you in no time.

"Component factors" isn't a well-defined term. The correct term would be "irreducible factors" to emphasize that those cannot be split further. The others are called reducible.

This is a theorem, which holds in some kind of number sets and which says: irreducible numbers are prime numbers. It is not automatically true, e.g. the real numbers don't have any primes.

A number is called prime, if whenever it divides a product, it has to divide (at least) one of the factors: ##p\,\vert \,a\cdot b \Longrightarrow p\,\vert \,a \, \vee \, p\,\vert \,b##.
This means that primes are always irreducible but not necessarily all irreducible elements are prime.

Yes, because it can be shown that irreducible and prime are the same in the integers. The fact that unities like ##\pm 1## are ruled out is deliberately made. The purpose to do so is to formulate theorems in a reasonable way, since one can always add arbitrary many ones or evenly many minus ones to a factorization without changing the result. If unities (elements with a multiplicative inverse) were allowed, it would be harder to get to the core meaning of prime numbers or unnecessarily complicated to formulate results: "Let ##p## be a prime, but not a unity, then ..." would have been to added almost everywhere.

I am a bit confused ,i was trying to learn polynomial factorization . but before that , i thought i would refresh my arithmetic factorization skills . these things are confusing me a lot .

Especially terms like , factors , prime factors ,factorize , factorization

number factors.
algebraic factors

not sure which term belongs to which or what term should be used where ...

If you write a polynomial as the product of two or more polynomials, you have factored the polynomial.

Here is an example

how do i factor a polynomial ?

The steps involved ...

anyway what my main aim is to learn about polynomial factorization itself , i hope that's what it is actually called ... :)

factors -- expressions that can be multiplied to produce some given expression. For example, 4 and 9 are factors of 36, because 4 * 9 = 36. Also 2x and 5x are factors of 10x^{2}, because (2x)(5x) = 10x^{2}. prime factors - expressions that can't be further broken up into simpler factors. For example, the prime factors of 36 are 2, 2, 3, and 3, since 2 * 2 * 3 * 3 = 36. 4 and 9 are factors of 36, but they aren't prime factors. Both 2 and 3 are prime numbers (or primes). A number (integer) is prime if its only factors are 1 and the number itself. factorize (or factor) - (verb) to reduce an expression into a product (multiplication) of subexpressions. In the US we typically say "factor an expression" rather than "factorize an expression." This shorthand usage might be confusing to some, because "factor" can be used as both a noun (e.g., "7 is a factor of 49") and as a verb (e.g., "factor 15 into a product of prime numbers.").
In your previous post you factored the trinomial ##6x^2 + 19x + 16## into a product of two binomial factors ##(2x + 3)(3x + 5)##. factorization - (noun) the action of reducing an expression into a product of its factors.

A little bit of an update is ok i guess ... This started with a lot of confusions about arithmetic and algebra .
I thought i would share some stuffs which i was able to narrow down to for beginners like me ...
I hope i am not going too much off topic here .

i found this really nice website when i was searching for some terms

@rosiekidcute, you posted images of all of those factoring examples. If you understand them all and are able to work examples of them correctly, then you're in pretty good shape, and should continue on with other topics. As I recall, you are reviewing topics in algebra from a book you purchased. I would suggest just continuing to work through your book. Possible topics would include working with various types of equations, such as equations with radicals, equations with rational expressions, equations with absolute values, linear and quadratic inequalities, graphs of linear functions, quadratic functions, and so on.