f(x)=x^3+7x-22 and linear factorization of f(x)
It is interesting that your title puts "finding zeros" first! Of course a typical way of solving a polynomial equation is to first factor it, but it can be done the other way. By the "rational root theorem", the only possible rational roots to this equation are $\displaystyle \pm1$, $\displaystyle \pm 2$, $\displaystyle \pm 11$, or $\displaystyle \pm 22$, the various factors of 22. By simply putting those numbers into f(x), it is easy to see that the only rational root is x= 2. That tells us that x-2 is a factor. Divide $\displaystyle x^3+ 7x- 22$ by x- 2 to find the other (quadratic) factor. You can solve that using the quadratic formula and use those roots to find the two other linear factors.