Let f(x) = `-x^4-2x^3+6x-4`. Solve

2 Answers

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sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

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` .

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jasminehems | Student, Undergraduate | (Level 1) Honors

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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:

`q: -1`

`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.