First, we could try to factorize the polynomial. We could group the first and third terms, and the second and fourth term:
[tex]\displaystyle{ (3x^4-27x^2)-(x^3-9x)=3x^2(x^2-9)-x(x^2-9)[/tex]
Now, x(x^2-9) is a common factor:
[tex]
\displaystyle{ x(3x-1)(x^2-9)[/tex].
By the difference of squares formula, our final factorization is :
x(3x-1)(x-3)(x+3).
Here we can see clearly that the roots of f(x) are 0, 1/3, 3, and -3.
Remark: we grouped the terms in pairs as we did, noticing that the ratio of the exponents, and coefficient in each pair was the same.
Answer: 0, 1/3, 3, and -3.
