How do you find the zeros of a function with 2 variables??

Question

2x⁴+(2+2b)x³+(-10+2b)x²+(2-12b)x-12

Answer 1

We need two equations if you need to solve for both x and b. If, however, b is a parameter and x is the independent variable, we can solve the problem as follows.

Collect the function in terms that depend on b and terms that do not, i.e. f(x,b) = 2x^4 + 2 x^3 - 10x^2 + 2x - 12 + (2b x^3 + 2b x^2 - 12bx)

 

This function factorizes as f(x,b) = 2*(x^2+bx+1)(x^2+x-6)

The zeros of the function require x^2+x-6 = 0 (i) or (x^2 + bx + 1) = 0 (ii)

(i) has solutions x = 2, x = -3

(ii) has solutions x = (-b ± sqrt(b^2-4))/2

 

Hence the four zeros of the function are x = 2, x= -3, x = (-b ± sqrt(b^2-4))/2. Note that the latter two zeros show a relation between x and b rather than specific values for x or b. The first two zeros however are independent of b and are thus solutions even for the two variable situation.

Answer 2

Two functions in two variable (a system) are needed to solve for zeros. You present only one function.