Solving Polynomial Inequalities Using Synthetic Division and the PQ Rule

[Wa1337]

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

 

[gb7nash]

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?

 

[Wa1337]

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 ?

 

[SammyS]

[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.

 

[Wa1337]

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.

 

[SammyS]

Use the rational root theorem (the PQ rule). Then use synthetic division.

 

[Wa1337]

Yeah I got 4x⁴-20x³+29x²-6x-9

 

[SammyS]

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 .

 

[Wa1337]

(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.