How can we factorize cubic polynomials?

e.g. y³+ 2y²+ y+ 1

In general, we can't! One way is to look for "zeroes" of the polynomial: if x= a makes the polynomial equal to 0, then (x- a) is a factor. But it is not easy to find zeroes and, generally, when we say "factor" we mean to factor with integer coefficients- and that may not be possible.

 

The best I can do is notice that we can factor "y" out of the first three terms: [tex]y(y^2+ 2y+ 1)+ 1[/tex] and that [tex]y^2+ 2y+ 1= (y+ 1)^2 so that [tex]y^3+ 2y^2+ y+ 1= y(y+ 1)^2+ 1[/tex] but that has not "factored" the original polynomial.