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In order to get check whether it is zero or not just substitute the zeros of polynoomial values in the equation
4x^3-5x^2-23x+6=0
x=(-2)
4(-2)^3-5(-2)^2-23(-2)+6=0
=4(-8)-5(4)+46+6
=-32-20+52
=-52+52=0
=== -2 is zero of the polynomial ====
4x^3-5x^2-23x+6=0
x=1/4
4(1/4)^3-5(1/4)^2-23(1/4)+6=0
4(1/64)-5(1/16)-23/4+6
1/16-5/16-23/4+6
LCM is 16
1-5-92+96/16
-97+97/16
0/16==0
1/4 is the zero of polynomial
Last thing to be answered that is 3 root for the equation
for this we need to divided any of the root we got with polynomial given.
x=-2
can be written as (x+2)
on dividing 4x^3-5x^2-23x+6 with x+2 the quotient is 4x^2-13x+3
on finding roots for it we get x=1/4 and x=3
therefore the three roots are 1/4,-2 and 3