- 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 three zeroes of a cubic polynomial $ x^{3}+a x^{2}-b x+c $ are positive, then at least one of $ a, b $ and $ c $ is non-negative.

Given:

If all three zeroes of a cubic polynomial \( x^{3}+a x^{2}-b x+c \) are positive, then at least one of \( a, b \) and \( c \) is non-negative.

To do:

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

Solution:

Let $\alpha, \beta$ and $ \gamma$ be the zeroes of cubic polynomial $x^{3}+a x^{2}-b x+c$

This implies,

Product of zeroes $=\alpha \beta \gamma=-\frac{\text { constant term }}{\text { Coefficient of } \mathrm{x}^{3}}$

$=\frac{-c}{1}$

$\alpha \beta \gamma=-c$

Given that, all three zeroes are positive.

This implies,

The product of all three zeroes is also positive.

$\alpha \beta \gamma>0$

$-c>0$

$c<0$

Sum of the zeroes $=\alpha+\beta+\gamma=-\frac{\text { coefficient of } \mathrm{x}^{2}}{\text { Coefficient of } \mathrm{x}^{3}}$

$=\frac{-a}{1}$

$=-a$

But $\alpha, \beta$ and $\gamma$ are all positive.

This implies, its sum is also positive.

$\alpha+\beta+\gamma>0$

$-a>0$

$a<0$

Sum of the product of two zeroes at a time $=\frac{\text { coefficient of } \mathrm{x}}{\text { Coefficient of } \mathrm{x}^{3}}$

$=\frac{-b}{1}$

$= -b$

Therefore, the cubic polynomial $x^{3}+a x^{2}-b x+c$ has all three zeroes which are positive only when all constants $a, b$ and $c$ are negative.

Hence, the given statement is false.

- Related Questions & Answers
- Sum of the Series 1 + x/1 + x^2/2 + x^3/3 + .. + x^n/n in C++
- Find minimum positive integer x such that a(x^2) + b(x) + c >= k in C++
- What are the differences in between python 2.x and python 3.x versions?
- Find maximum among x^(y^2) or y^(x^2) where x and y are given in C++
- Differences between Python 2.x and Python 3.x?
- Program to find sum of 1 + x/2! + x^2/3! +…+x^n/(n+1)! in C++
- Minimum positive integer value possible of X for given A and B in X = P*A + Q*B in C++
- What are the key differences between Python 2.7.x and Python 3.x?
- Count of all possible values of X such that A % X = B in C++
- Construct SLR (1) parsing table for the following grammar S → x A y |x B y |x A z A → q s | q B → q
- Important differences between Python 2.x and Python 3.x with examples
- Are true and false keywords in java?
- Evaluate a 2-D polynomial at points (x, y) in Python
- Java Program to Check if two of three Boolean variables are true
- Evaluate a 3-D polynomial at points (x, y, z) in Python
- Evaluate a 2-D polynomial on the Cartesian product of x and y in Python