Let g(x) = (x - 1)5(x + 1)6(x - 2)7. What is the multiplicity of the zero x = 1?

Answer 1

Do you mean
g(x) = (x-1)^5*(x+1)^6*(x-2)^7 ?

The multiplicity of the x=1 root is 5.

See http://people.richland.edu/james/lecture/m116/polynomials/polynomials.html
for more details

Answer 2

Assuming those numbers following the factors are exponents, then the exponents are the multiplicities.

(x-1)^3 = (x-1)(x-1)(x-1), so 1 is a root 3 times.