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### Prime Expressions

An algebraic expression is called 'prime' when it can not be factored. For example, 5x + 1 is a prime expression because there is no common factor between the two terms '5x' and '1' which can be factored out.

We know that a prime number has only two factors - 1 and itself. Similarly a prime expression has only two factors - 1 and itself. No other number, term or expression is its factor.

Conversely, 5x + 10 is not a prime expression because the two terms in it, '5x' and '10' have a common factor 5, which can be factored out of the expression, thus making it 5(x + 2).

Linear expressions (the ones in the examples above) can be factored only by one method: factoring out a common factor, but quadratic and higher degree expressions can be factored by various other methods. Thus, in order to test whether a higher degree expression is prime or not, you need to try to factor it in different ways.

A sure-fire signal that an expression (quadratic and higher degree) can not be factored is when its graph does not have any x-intercepts. This is a direct result of the factor theorem.

More on Prime and Non Prime Quadratic Expressions