Application

Suppose x R (the reals) is a solution of the equation

x⁵ - 8x⁴ + 16x³ + 3x² - 14x = 4.

Using more advanced methods than treated so far, real polynomials can be factored. Bringing 4 to the left hand side and factoring, we find

(x² - 3x - 1) (x³ - 5x² + 2x + 4) = 0.

Since R is a domain, we conclude

x² - 3x = 1    or    x³ - 5x² + 2x = -4.

So x is a solution of one of two equations of smaller degree.