When we talk about factoring expressions, we’re referring to the process of breaking down a polynomial into simpler components, or factors, that when multiplied together yield the original expression. However, some expressions cannot be factored using real numbers or standard algebraic methods.
For example, the expression x² + 1 does not factor over the real number system. This is because there are no two real numbers that multiply together to produce a positive number (in this case, +1) while also summing up to zero. The quadratic formula confirms this, as the discriminant (b² – 4ac) for this expression is negative, leading to complex roots.
In contrast, expressions like x² – 1 can easily be factored into (x – 1)(x + 1), demonstrating that not all expressions can be broken down in this way. Identifying expressions that do not factor is a critical skill in algebra, allowing one to understand the nature of the polynomial and the solutions it might yield.