Question

Solve the polynomial inequality. $$x^{3}+5 x^{2}+4 x<0$$

   Solve the polynomial inequality.
$$x^{3}+5 x^{2}+4 x<0$$

Precalculus : Building Concepts and Connections

Precalculus : Building Concepts and Connections

Revathi Narasimhan 1st Edition

Chapter 3, Problem 13

Instant Answer

Solved by Expert Ramzi Deek

05/05/2020

Step 1

We can factor out an $x$ from each term to get: $$x(x^{2}+5x+4)<0$$ Then, we can factor the quadratic expression to get: $$x(x+4)(x+1)<0$$ Show more…

Show all steps

lock

Thanks for your feedback!

Solve the polynomial inequality. $$x^{3}+5 x^{2}+4 x<0$$

Play audio

Feedback

Powered by NumerAI

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Polynomial Factorization

This is the process of expressing a polynomial as a product of its simpler factors, typically lower-degree polynomials. Factorization helps in breaking down complex expressions into more manageable pieces, enabling easier identification of the zeros or roots, which are essential in further analysis such as solving equations or inequalities.

Critical Points

Critical points are the values of the variable that make the polynomial equal to zero. These points divide the real number line into distinct intervals where the sign of the polynomial may change. Identifying the zeros is a crucial step in solving polynomial inequalities since they indicate where the function can switch from positive to negative or vice versa.

Sign Analysis

Sign analysis involves testing the polynomial in the intervals determined by the critical points to determine whether the polynomial's value is positive or negative. By analyzing the sign in each interval, one can identify where the inequality holds true. This systematic method is central for solving polynomial inequalities as it helps to precisely define the solution set.

Recommended Videos

Transcript

Ace is your personal tutor. It breaks down any question with clear steps so you can learn.

Start Using Ace

Continue solving this problem

Create a free account to:

View full step-by-step solution
Ask follow-up questions with Ace AI
Save progress and study later