Question

given that 1 is a zero of p(x)=3x^4-x^3-8x^2+2x+4, factor the polynomial completely, and then list all its zeros.

Answer 1

Divide p over (x-1) with euclidian division we get the quotient:
$3x^3+2x^2-6x-4$
Note that -2/3 is also roots of the above polynomial. 
Checking:
$3(-2/3)^3+2(-2/3)^2-6(-2/3)-4=0$
Dividing again with euclidian algorithm we get the quotient:
$3x^3+2x^2-6x-4$ over $3x+2$ is $X^2-2$.
Final factorization:
$p(x)=(x-1)(3x+2)(x-\sqrt2)(x+\sqrt2)$