Find the zeros: Includes a cube

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Homework Help Overview

The discussion revolves around finding the zeros of a cubic function, specifically f(x) = 4x^3 - 24x^2 - x + 6. Participants are exploring algebraic methods to identify the roots of the polynomial.

Discussion Character

  • Exploratory, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • Participants discuss factoring the polynomial and consider the implications of the constant term. There are attempts to identify rational roots and clarify the factors derived from the polynomial.

Discussion Status

The conversation includes various interpretations of the factors and roots. Some participants confirm the identified zeros while others seek clarification on the factorization process. There is an ongoing exploration of the correctness of the proposed solutions.

Contextual Notes

Participants are working under the constraints of algebraic methods and are questioning the accuracy of their factorization and root identification. There is an emphasis on verifying results through substitution into the original polynomial.

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Homework Statement



Find the zeros of the function algebraically.

Homework Equations



f(x) = 4x^3 - 24x^2 - x + 6

The Attempt at a Solution



If all quantities had an x in them, I'd just factor out and x, and treat it as a quadratic. But that freaking 6 is ruining my plan and I'm stuck.
 
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You'll have to try and factor it. Look for a rational root.
 
k, so I factored it to (1 - 4x^2)(-x - 6)

...what are the zeros? 6? Someone help me interpret that...pleeease.
 
How did you get the (-x-6) factor? I get just (x-6). Or did you mean to write (-x+6)? Then to find the roots, just set each factor equal to zero, right? The product of things can only be zero if one of the things is zero.
 
Damnit, yeah I meant to have (x - 6).
 
Okay so they're 6, +1/2, -1/2. Correct if I'm wrong. If not, thanks.
 
They are correct. Didn't want to just leave you hanging. You can also check for yourself, just put those values into the polynomial and see if you get zero.
 

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