Max Value of k for Cubic Polynomial Factoring

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Homework Help Overview

The discussion revolves around determining the largest value of k for which two given quadratic polynomials, Q_1(x) and Q_2(x), are factors of an unspecified cubic polynomial P(x).

Discussion Character

  • Conceptual clarification, Assumption checking

Approaches and Questions Raised

  • Participants express confusion regarding the necessity of knowing the specific form of P(x) to determine the factors. Some suggest that a general cubic polynomial could suffice for analysis, while others question the implications of k's value on the existence of common factors across all cubic polynomials.

Discussion Status

The discussion is exploring different interpretations of the problem statement. Some participants are attempting to clarify the assumptions about P(x) and how they relate to the factors Q_1 and Q_2, while others are questioning the validity of the problem's premise.

Contextual Notes

There is a noted ambiguity in the problem statement regarding the relationship between k and the cubic polynomial P(x), leading to varied interpretations among participants.

ehrenfest
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Homework Statement


The polynomial P(x) is cubic. What is the largest value of k for which the polynomials Q_1(x) = x^2+(k-29)x-k and Q_2(x) = 2x^2+(2k-43)x+k are both factors of P(x)?

Homework Equations


The Attempt at a Solution


I don't understand the question. How can you determine whether Q_1 and Q_2 are factors of P(x) when they do not tell you what P(x) is!?
 
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Presumably you only need to know that it is cubic, so take some general cubic function.
 
How could the answer possibly not depend on what P(x) is? If not, and the answer is greater than 1, that implies that ALL cubic polynomials have a common factor which is absurd!
 
I agree - the wording could have been better. I think what they want you to do is assume that Q_1 and Q_2 are divisors of P, find the values of k for which this is possible, and give them the largest of these values. Using this interpretation, all you need to know about P is that it's a cubic.
 

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