Let f(x) = `-x^4-2x^3+6x-4`. Solve
Since the question was tagged with calculus and since it has not indicated what needs to be solved, I assume that you may need to evaluate the first order derivative, hence, you need to differentiate the function with respect to x, such that:
`f'(x) = (-x^4 - 2x^3 + 6x - 4)' => f'(x) = -4x^3 - 6x^2 + 6`
Hence, evaluating the first order derivative of the given polynomial function, yields `f'(x) = -4x^3 - 6x^2 + 6` .
This polynomial can be solved using the Rational Root Theorem.
Firstly, you must find `p/q,` where `q` is all divisors of the leading coefficient, and `p ` is all divisors of the constant. So, for this quintic polynomial:
`p: -4; +-1, +-2, +-4`
`p/q: ``1/-1 = +-1; 2/-1 = +-2; 4/-1 = -+4`
`Roots: +-1, +-2, +-4`
Now you can run these roots through synthetic division to find the one that does not have a remainder.
The one that does not have a remainder goes into the regular factoring form, `(x+-l)(x+-m)(x+-n).` Then you solve for x, and you will have your zeroes.
If you need a reminder about how to do synthetic division, the link below should help.