Factor: x^4 + x^3 - 4x^2 + x +1 | Olympiad Question

We introduce two methods to solve this problem. First, we find a root using the formula for the relationship of the roots to the polynomial coefficients, and then factorize using long division. Repeating the same idea, we got the answer. The second method is unusual. The idea is based on the completing the square method.

[rorydaulton]

You factored the polynomial as far as possible using integer or rational coefficients. But you did not give that requirement at the start of the video, so the usual assumption is that you mean to factor over the reals or perhaps the complex numbers. That last factor of yours, x² + 3x + 1 can be factored further over the reals. So the full factorization of your polynomial over the real or the complex numbers is

(x - 1)² · (x+ (3 - sqrt(5))/2) · (x + (3 + sqrt(5))/2)

[桜木秋水]

fainal answer?

(1)if Coefficient is integer
(x-1)^2(x^2+3x+1)

(2)in real range
x^2 + 3x + 1=0
x = (-3±√5)/2