Algebra 2
posted by kaley on .
find all the zeros of the polynomial function
F(x)= x^4+4x^36x^236x27

There are websites that solve such rootfinding questions automatically, but they are not very instructive.
If there are integer roots, they must be even divisors of 27: +/ 1, 3, 9 or 27.
That is a consequence of the "rational roots theorem", which you should learn.
One such root is 1, so x+1 is a factor.
The other factor is
(x^4+4x^36x^236x27)/(x+1)
= x^3 +3x^2 9x 27
(obtained with polynomial long division)
x = 3 is clearly another root, so (x3) is another factor. Divide the cubic by (x3) and you get
x^2 + 6x +9 = 0
which can be factored to give
(x+3)^2 = 0
That means x = 3 is a double root.
The roots are 3, +3 and 1