Solving Polynomial Inequalities Using Synthetic Division and the PQ Rule

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Homework Help Overview

The discussion revolves around solving a polynomial inequality involving a fifth-degree polynomial. Participants are exploring methods to find the zeros of the polynomial and analyze its behavior in relation to the inequality.

Discussion Character

  • Exploratory, Problem interpretation, Assumption checking

Approaches and Questions Raised

  • Participants discuss the use of synthetic division and the PQ rule to find zeros. There is mention of the rational root theorem and the process of testing values between identified zeros. Some participants express uncertainty about the initial polynomial and its roots.

Discussion Status

There is ongoing exploration of the polynomial's roots, with some participants confirming the polynomial's correctness and others discussing the outcomes of synthetic division. Guidance has been offered regarding the rational root theorem and the next steps in the problem-solving process.

Contextual Notes

Participants note constraints such as the prohibition of calculator use and the need to clarify the polynomial's roots, which initially led to confusion. There is an emphasis on ensuring the polynomial is correctly stated for the context of the homework assignment.

Wa1337
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Homework Statement


4x5-16x4+9x3+23x2-15x-9 > 0


Homework Equations


Synthetic division
PQ Rule?


The Attempt at a Solution


Don't know how or where to begin
 
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The goal here is to find the zeros. Once you find all zeros, you can plug in an arbitrary value between each zero. Do you know the rational root theorem?
 
gb7nash said:
The goal here is to find the zeros. Once you find all zeros, you can plug in an arbitrary value between each zero. Do you know the rational root theorem?

The Pq rule ?
 
[STRIKE]There are no rational roots. There are 3 real roots & 2 complex roots which aren't real.

Are you sure this is the correct polynomial?[/STRIKE]

Added in Edit:

Never mind. (I cut & pasted it incorrectly! DUH)

All 5 roots are real. 3 of them are rational.
 
Last edited:
Yes this is the correct polynomial given to us for our test review.

REVISED: Okay, well I used synthetic division with the whole polynomial and used the -1 as a factor and it worked well, but now I have 4x4-20x3+29x2-6x-9. What should I do to get further and am I going in the right direction? Thanks.
 
Last edited:
Use the rational root theorem (the PQ rule). Then use synthetic division.
 
Yeah I got 4x4-20x3+29x2-6x-9
 
Wa1337 said:
Yeah I got 4x4-20x3+29x2-6x-9

So 4x5-16x4+9x3+23x2-15x-9 = (4x4-20x3+29x2-6x-9)(What) ?

What root did you find? There are two other rational roots & 2 irrational roots. Keep working at it.

Try graphing y = 4x4-20x3+29x2-6x-9 .
 
(4x4-20x3+29x2-6x-9)(x-1) is the rest of the equation and ok I'll update you when I continue.

We may not use a calculator.
 

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