Rational root theorem states that if a polynomial a_0x^n+a_1x^(n-1)+a_2x^(n-2)+...+a_n, has rational roots p/q, where p and q are integers, then q is a factor of a_0 and p is a factor of a_n.
Hence to find all possible zeros of f(x)=3x^5−2x^4−15x^3+10x^2+12x−8,
we have to find roots of the equation 3x^5−2x^4−15x^3+10x^2+12x−8=0
For this we start by factors of constant term -8 such as {1,-1,2,-2,4,-4,8,-8}.
It is observed that x=1 makes f(x)=0 and hence (x-1) is a factor of f(x) as f(1)=3-2-15+10+12-8=0.
Similarly for x=-1 makes f(x)=0 and hence (x+1) is a factor of f(x) as f(1)=-3-2+15+10-12-8=0.
x=-2 makes f(x)=0 as f(2)=3*32-2*16-15*8+10*4+12*2-8=96-32-120+40+24-8=0 and hence (x-2) is a factor of f(x).
x=--2 makes f(x)=0 as f(2)=3*(-32)-2*16-15*(-8)+10*4+12*(-2)-8=-96-32+120+40-24-8=0 and hence (x+2) is a factor of f(x).
As four factors of f(x) have been found, we can have fifth factor by dividing f(x) by these and result would be (3x-2) and hence (3x-2)=0 or x=2/3 is another zero of f(x)
Hence, zeros of f(x) are {-2, -1, 1, 2/3. 2}
