Polynomial Equations

Polynomial Equations in Detail


A polynomial equation is one of the foundational concepts of algebra in Mathematics. Having a clear and logical sense of how to solve a polynomial problem will allow students to be much more efficient in their examinations and will also act as a firm base in their higher studies. Polynomial equations are in the forms of numbers and variables. A polynomial equation is a form of an algebraic equation. There is a minute difference between a polynomial and a polynomial equation. Polynomials are expressions whereas polynomial equations are expressions equated to zero.


A polynomial can be expressed as the sum of a finite number of terms, each term being the product of a constant coefficient and one or more variables raised to a positive integer exponent (or power).


It is an expression that can only have non-negative integer exponents (or power) and the operations of addition, subtraction, and multiplication. Simply put, it is a polynomial only if the expression can be written without division. Also, an expression is not classified as a polynomial if even one of the terms has a negative exponent (or power). An expression is not classified as a polynomial if even one of the terms has a fraction exponent (or power).


An equation is an assertion that two expressions are equal. This is shown by writing an 'equal to' sign denoted by '=' between the two expressions. It is an assertion from which a particular quantity is supposed to be determined.


A polynomial Equation is simply a polynomial that has been set equal to zero in an equation.


We will try to understand polynomial equations in detail. We will learn about the degree of a polynomial, types of a polynomial equation and most importantly, how to solve a polynomial equation.


Vedantu provides study material required to solve quadratic equations which is one of the most commonly asked questions in Mathematics examinations. Students can download it for free from the Vedantu website and app.


What is a Polynomial Equation?

A polynomial equation is an expression consisting of variables, coefficients and exponents. A polynomial function is one which has a single independent variable. The exponents in a polynomial equation can only be in the form of positive integers, therefore, any negative integer exponent is disqualified as a polynomial equation.


Degree of Polynomial

The independent variable can occur multiple times in a polynomial. These different occurrences of the variable are separated by operations of addition, subtraction and multiplication. The degree of the polynomial is defined as the highest degree of the exponent that exists in the equation. It is also called the order of the polynomial equation.


For the polynomial x2 + 3x + 6 , the degree or the order of the polynomial is 2.


Polynomial Formula

A polynomial is generally of the form anxn. Here, a is called the coefficient, x is the independent variable and n is the exponent. Equating this polynomial to zero gives us a polynomial equation. The value of the exponent n can only be a positive integer as discussed above.


Any polynomial function can be of the form

F(x) = anxn + an-1xn-1 + an-2xn-2 + . . . + a1x + a1 = 0


is the general formula of a polynomial.

On putting the values of a and n, we will obtain a polynomial function of degree n.

F(x)=2x2+5x=0

Here, the polynomial 2x2 + 5x, equated to zero gives us the polynomial equation F(x) = 2x2 + 5x = 0 with degree 2.


Types of Polynomial Equation

Polynomial equations are classified upon the degree of the polynomial. For practical reasons, we distinguish polynomial equations into four types.


  1. Monomial/Linear Equation

A polynomial equation with only one variable term is called a monomial equation. It is also called a linear equation. The algebraic form of a linear equation is of the form:

ax + b=0, where a is the coefficient, b is the constant and the degree of the polynomial is 1.

Examples:

2x + 10 = 0

x - 5 = 0


  1. Binomial/Quadratic Equation

A polynomial with two variable terms is called a binomial equation. It can also be called a quadratic equation. The algebraic form of a quadratic equation is of the form:

ax2 + bx + c = 0, where a and b are coefficients, c is the constant and the degree of the polynomial is 2.

Examples:

2x2 + 2x + 2 = 0

x2 - 4=0


  1. Trinomial/Cubic Equation

A polynomial with three variable terms is called a trinomial equation. It is also called a cubic equation. The algebraic form of a quadratic equation is of the form:

ax3 + bx2 + cx + d = 0, where a, b and c are coefficients, d is the constant and the degree of the polynomial is 3.

Examples:

x3 + 2x2 + 3x - 5 = 0

2x3 - 5x = 0


  1. Polynomial Equation

A polynomial with more than three variable terms is called a polynomial equation. It is of the form

anxn + an-1xn-1 + an-2xn-2 + . . . + a1x + a1 = 0.


Examples,

4x4 + 2x3 + x2 + 5 = 0

10x5 + 2x - 10 = 0


Solving Polynomial Equations

Polynomial equations are generally solved with the hit and trial method. We put in the value of the independent variable and try to get the value of the expression equal to zero.


In case of a linear equation, obtaining the value of the independent variable is simple. We solve the equation for the value of zero.


For the polynomial, 2x - 4 = 0

2x = 4

x = 2


However, this solution is not easily applicable in higher degrees of polynomials, therefore, we go with the hit and trial method.


Application of Polynomial Equations in Real Life

Polynomials, polynomial equations and polynomial functions are used to graph curves in the real world. Professionals from various careers who carry out complex calculations to determine the nature of curves in real-life scenarios utilise polynomial equations and functions.

 

Civil Engineering requires their use to determine the curvature of roads, flyovers, bridges and other complex spatial structures. Global Positioning System (GPS) calculates a complex system of polynomial equations to locate your position. 


Polynomial functions prove to be important mechanisms to predict traffic patterns to gauge your estimated time of arrival to your given destination on your public transport apps like Uber and Ola. These equations allow governments to efficiently implement traffic control measures like planning the correct placement of traffic lights to ensure smooth roads.


Robotics employs polynomial equations to predict the trajectory of projectiles. These equations and functions are instrumental in computational biochemistry to model interacting proteins. 


The banking industry and accountants use them for tax and loan calculations. Economists and businessmen use polynomial equations to model market patterns and cost analysis. Statisticians, data analysers and researchers also use them among many other expressions in Mathematics.


People aspiring to be in the nursing industry are also expected to know polynomial equations. They use it to monitor parameters like the depleting or increasing presence of a drug in small quantities in the bloodstream of the patient and the medication dosage required in accordance with the changing health status over time in the hospital.


Even environmental conservationists, nature enthusiasts, the weather department and the forestry department put polynomial functions to good use to predict how natural resources are changing (increasing or depleting) over a period of time-based on the previous data gathered earlier. 


Damage Control teams and Crisis Management teams use them to monitor the curving trajectory of cyclones and hurricanes on the basis of a bunch of variables like wind speed.


To summarise, polynomials, polynomial equations and polynomial functions may seem like Mathematical jargon on paper, but in reality, they are used in most scientific applications. They are crucial for the success of entire industries that help science make human lives more advanced, efficient and sustainable.

FAQs on Polynomial Equations

1. How do we Solve a Quadratic Polynomial Formula?

One method to solve a quadratic formula is to use the hit and trial method, where we put in different values for the independent variable and try to get the value of the expression equal to zero. However, this method of hit and trial can be tiresome, so we try to find the roots of the equation using the quadratic formula.

x = [-b ± √(b2 - 4ac)]/2a


A quadratic equation is of the form of ax2 + bx + c = 0, where a and b are coefficients and the degree of the equation is 2, which means that there are two roots of the equation

x = [-b ± √(b2 - 4ac)]/2

Using the quadratic formula, we obtain the roots of the equations instantly.

2. Solve the Following Polynomial Equation, 5x2 + 6x + 1 = 0.

The above equation is a polynomial equation with degree 2. It is a quadratic equation with two roots.


The equation 5x2 + 6x + 1 = 0 is a quadratic equation, where a,b and c are real numbers.

a = 5

b = 6

c = 1


So, using the quadratic formula,

x = [-b ± √(b2 - 4ac)]/2

x = [-6 ± √(62 - 4 x 5 x 1)]/2

x = −6±√(36−20)/2

x = −6±√(16)/2

x = −6±4/2

The values of x are,

x = -2/2 = -1

x = -10/2 = -5

The roots of the equation are -1 and -5. So the values of x that satisfy the equation are -1 and -5.

3. Are radicals a part of polynomials?

No. A polynomial must not contain square roots of any variable. Therefore, a polynomial cannot have radicals.

Students can download the study material for free from the Vedantu website and app where they can revise the properties of a polynomial for exam preparation.

4. How to classify a polynomial?

When a polynomial is written in standard form, all the terms in it are arranged from the highest degree to the lowest degree. The degree of a polynomial is the largest exponent on one of its variables (for a single variable). The following names are assigned to polynomials on the basis of their degree:

  • When the degree is 0, the polynomial is called zero or constant polynomial.

  • When the degree is 1, the polynomial is called a linear or monomial polynomial.

  • When the degree is 2, the polynomial is called a quadratic or binomial polynomial.

  • When the degree is 3, the polynomial is called a cubic or trinomial polynomial.

5. How to write a polynomial in standard form?

Writing a polynomial in the standard form helps identify the degree of a polynomial and classify it. The following are the easy steps to write a polynomial in standard form:

  • Note all the terms.

  • Group all the like terms together.

  • Note the term with the highest exponent first.

  • Queue the remaining terms in descending order of exponents.

  • Note the constant term (the term without a variable) towards the end.

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