Solving Polynomial Inequalities Using Synthetic Division and the PQ Rule

In summary, the conversation is about solving a polynomial inequality and using synthetic division and the rational root theorem to find the roots. The polynomial given has 3 real roots and 2 complex roots, but after correcting a mistake in the original polynomial, it is found that all 5 roots are real. The conversation also touches on using a graph to check the roots and the use of a calculator during the test.
  • #1
Wa1337
33
0

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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  • #2
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?
 
  • #3
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 ?
 
  • #4
[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:
  • #5
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:
  • #6
Use the rational root theorem (the PQ rule). Then use synthetic division.
 
  • #7
Yeah I got 4x4-20x3+29x2-6x-9
 
  • #8
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 .
 
  • #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.
 

1. What is a polynomial inequality?

A polynomial inequality is an inequality that contains one or more polynomials. A polynomial is a mathematical expression consisting of variables, coefficients, and exponents that are combined using addition, subtraction, multiplication, and non-negative integer exponents.

2. How do you solve a polynomial inequality?

To solve a polynomial inequality, you can follow these steps:

  1. Move all terms to one side of the inequality, leaving a 0 on the other side.
  2. Factor the polynomial to find its roots.
  3. Plot the roots on a number line and determine the intervals where the polynomial is positive or negative.
  4. Test a value from each interval in the original inequality to determine the solution set.

3. What is the solution set of a polynomial inequality?

The solution set of a polynomial inequality is the set of all values that make the inequality true. It can be represented using interval notation or set notation.

4. How do you graph a polynomial inequality?

To graph a polynomial inequality, you can follow these steps:

  1. Plot the roots of the polynomial on a number line.
  2. Use test points from each interval to determine the direction of the graph (up or down).
  3. Plot the graph accordingly, making sure to include any boundary points.
  4. Shade the region above or below the graph, depending on the direction of the inequality.

5. What are some real-world applications of polynomial inequalities?

Polynomial inequalities can be used to model and solve various real-world problems, such as determining the range of possible values for a variable in a given situation. For example, they can be used to find the maximum or minimum production levels for a company based on certain constraints, or to determine the range of possible values for a person's income based on their education level and job experience.

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