Descartes’ Rule of Signs for Quadratic Polynomials

The particular pattern of the representation of the relation between the variables and numbers is recognised as the polynomial equations in the terminology of the algebraic expression of certain kinds of equations. On the other hand, it has been seen that in algebra, Descartes’ rule of signs is used in order to find the numerical value of the zero that exists in the polynomial quadratic equation. Based on these two primary concepts, the current study will discuss the coefficients, graphs and formulas as well. 

Quadratic Polynomials: Overview

A quadratic polynomial is considered a polynomial equation that has the second degree as the highest monomial degree. As per this scenario, a polynomial quadratic equation is also known by the name of the second-order polynomial. This fact represents that in the equation of a quadratic polynomial, one of the variables has the requirement of being raised at the power of two. On the contrary, the powers of other variables need to be less than or equivalent to two. The main concern needs to be focused on not using the constant term or coefficients while going a rise to the quadratic function in polynomial equations. 

Explanation of Descartes’ Rule of Signs

The statement of the Descartes’ rule of signs is explained in the below section:

• As per the condition, the number of positive real roots needs to be equivalent to the changing numbers in the signs that lied between two coefficients that are consecutive to each other  
• The number of real roots that are positive needs to be lesser than the two consecutive coefficients by an even number
• The number of real roots that are negative is weather equivalent to the changes in sign of f(-x) or less than it by an even number

The Coefficient of Quadratic Polynomial Sum and Product of Roots

A relationship can be established between the coefficients and the roots of the quadratic equation by the usage of the roots of the equation that contains the features of quadratic polynomials. In order to determine the coefficients, the product and the sum of the roots of the polynomial quadratic equations need to be used. For an instance, if the roots are β and α, the sum of the coefficients and the roots will be x2 – (α + β) x + α. β = 0.

Quadratic Polynomial Graph

In a singular variable, a parabola gives the graph of the quadratic polynomial. As a good example, if the quadratic polynomial denotes the equation such as x2 + bx + c, based on this equation determined by the parabola will be y = ax2 + bx + c. in reference to such scenarios, for obtaining the quadratic polynomial graph, the substitution of the value of x is required for determining test points in the aforementioned equation. This particular scenario will be helpful in getting the corresponding value of y as well. 

Quadratic Polynomial Formula

In accordance with the algebra, it can be stated that the normal formula of a quadratic polynomial with a single variable is represented as ax2 + bx + c. in the background of this particular scenario, a quadratic polynomial equation has been represented as ax2 + bx + c=0. In order to get the solution, different methods have been used along with different formulas. Among these methods, the most simple formula is x = −b±√(b2−4ac) / 2a.

Notes on Quadratic Polynomial

As per the algebra, the quadratic polynomial has been considered as a polynomial that has a degree that is equivalent to two. In addition, two roots can exist in the representation of a quadratic polynomial. The discriminant that shares a value that is equal to b2 – 4ac has been used in order to check the nature of the roots of the quadratic equation. The sum of the roots and coefficients can be expressed as x2-(sum of the roots)x+ (Product of the roots). 

Conclusion

In order to sum up all that has been stated so far in the study, it has been found that in algebra, the determination of the number of positive roots of a polynomial equation, Descartes’ rule of signs is used. In this study, it has come to the forefront that, Descartes’ rule of signs deals with the determination of not only the positive roots but also the negative ones as well. This study has further displayed the coefficients of the polynomials in the context of quadratic polynomial sums and products of the root.