How do you know when you have completely factored a polynomial?

Hint: A polynomial equation is used to represent a function such that the function is defined in terms of the unknown variable. Some questions require us to find out the values of the unknown quantity at which the value of the functions comes out to be zero, those values of the unknown variable are called the roots of the polynomial equation.

Complete step-by-step answer:
If the degree of the equation is greater than 2, we convert the given equation into a polynomial of lesser degree by finding a root of that equation by hit and trial method and then dividing the original equation by the factor. This way we can decrease the degree of the polynomial and find its roots one by one. If the polynomial is already of a degree 2 then we can solve it simply by any of the methods like factorization, completing the square, quadratic formula, etc.
Thus, we know that we have completely factored an equation when no more roots can be obtained and the number of roots obtained is equal to the degree of the polynomial equation.

Note: An algebraic expression contains numerical values and alphabets, the alphabets represent some unknown variable quantities. An algebraic expression is said to be a polynomial equation when the unknown variables are raised to some power and the power is a non-negative integer. We know that a polynomial equation has exactly as many roots as the degree of the equation so we first observe the degree of the equation.