- Trending Categories
- Data Structure
- Networking
- RDBMS
- Operating System
- Java
- iOS
- HTML
- CSS
- Android
- Python
- C Programming
- C++
- C#
- MongoDB
- MySQL
- Javascript
- PHP

- Selected Reading
- UPSC IAS Exams Notes
- Developer's Best Practices
- Questions and Answers
- Effective Resume Writing
- HR Interview Questions
- Computer Glossary
- Who is Who

# Are the following statements 'True' or 'False'? Justify your answers.

If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign.

Given:

If all the zeroes of a cubic polynomial are negative, then all the coefficients and the constant term of the polynomial have the same sign.

To do:

We have to find whether the given statement is true or false.

Solution:

Let $p(x)=x^{3}+a x^{2}+b x+c$ is a cubic polynomial and $\alpha, \beta, \gamma$ be the roots of $p(x)$.

This implies,

Sum of the roots $=\alpha+\beta+\gamma=-a$

Sum of negative numbers is negative.

This implies,

$a$ is positive.

Product of the roots taken two at a time $=\alpha \cdot \beta+\alpha \cdot \gamma+\gamma \cdot \beta=b$

Product of two negative numbers is positive and sum of positive numbers is positive.

This implies,

$b$ is positive

Product of the roots $=\alpha \beta \gamma=-c$

Product of three negative numbers is negative

This implies,

$c$ is positive.

Therefore, the sign of all three coefficients will be positive.

Hence, the given statement is true.

- Related Questions & Answers
- Python Program to Compute a Polynomial Equation given that the Coefficients of the Polynomial are stored in a List
- Check if all elements of the array are palindrome or not in Python
- Python Pandas - Return whether all elements in the index are True
- Return True if all entries of two arrays are equal in Numpy
- Check if all the 1s in a binary string are equidistant or not in Python
- Are true and false keywords in java?
- Evaluate a polynomial when coefficients are multi-dimensional in Python
- Return true or false in a MySQL select if another field contains a string?
- MongoDB query to find on the basis of true or false values
- Check whether the frequencies of all the characters in a string are prime or not in Python
- How to determine if all elements are the same in a Java List?
- Python - Check if all the values in a list are less than a given value
- What are False Positives and True Positives in Cybersecurity?
- How to check if all the values in a numpy array are non-zero?
- Check if all bits of a number are set in Python
- JavaScript Checking if all the elements are same in an array