Finding Third Root & Values of p & q

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Homework Help Overview

The problem involves finding the third root and the values of p and q for the cubic equation 2x^3 + px^2 + qx - 4 = 0, given that 2 is a repeated root.

Discussion Character

  • Exploratory, Assumption checking, Conceptual clarification

Approaches and Questions Raised

  • The original poster attempts to substitute the known root into the equation to express q in terms of p but encounters a cancellation issue. Some participants suggest considering the implications of repeated roots and the role of derivatives. Others propose a factorization approach to relate the polynomial to its roots.

Discussion Status

Participants are exploring different interpretations of the problem, particularly regarding the nature of repeated roots and their implications for the polynomial's structure. Guidance has been offered regarding the factorization of the polynomial, but no consensus has been reached on the specific values of p and q or the third root.

Contextual Notes

There is a mention of the original poster's background in precalculus, which may influence their understanding of derivatives and polynomial behavior. The discussion also reflects uncertainty about the assumptions underlying the problem setup.

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Homework Statement


Given that 2 is a repeated root of the equation 2x^3 + px^2 + qx -4 = 0, find the third root and the values of p and q.


Homework Equations


b^2 - 4ac(?)


The Attempt at a Solution


Since it said 2 is a root I plugged in 2 as the value of x and rearranged the equation in terms of q, which I figured out to be q = -2(3+p), but when I plug that back into the original equation to try and find p, I got 16 + 4p - 12 - 4p - 4 = 0. The p's cancel out so I don't know if I think I've done something wrong.
 
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If you know about derivatives (which I assume you don't, since you posted in precalc): what do you know about the derivative if the original polynomial has a repeated root??

If you do not know about derivatives: note that if 2 is a repeated root, then your original polynomial can be written as 2(x-2)^2(x-c) for a certain c. Work that out and compare it to the original polynomial.
 
micromass said:
If you know about derivatives (which I assume you don't, since you posted in precalc): what do you know about the derivative if the original polynomial has a repeated root??

If you do not know about derivatives: note that if 2 is a repeated root, then your original polynomial can be written as 2(x-2)^2(x-c) for a certain c. Work that out and compare it to the original polynomial.
The OP posted in the calculus section, also, but I deleted it there. This seems to me to be a problem that doesn't require calculus.
 
micromass said:
If you know about derivatives (which I assume you don't, since you posted in precalc): what do you know about the derivative if the original polynomial has a repeated root??

If you do not know about derivatives: note that if 2 is a repeated root, then your original polynomial can be written as 2(x-2)^2(x-c) for a certain c. Work that out and compare it to the original polynomial.

Where does c come from and what does "certain c" mean exactly?
 
c is the third root that you're trying to find. "For certain c" means that we don't know exactly what that number is, but we know that the 3rd degree polynomial with a repeated root of 2 has to factor into 2(x - 2)2(x - c) for some real number c.
 

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