Roots of Polynomial Equation


Polynomial is a chapter that we study in the Algebra part of our Mathematics curriculum. You are familiar with terms such as variables and exponents. You have been in touch with algebraic expressions such as 3g x 4h or 120k or to be more accurate, 4a + 3b -5c etc. Polynomials are the sum of these variables and exponents. You must have also heard of ‘term’. It is the individual part of the expression. Polynomials are expressions which contain more than two or three terms. In this article, we see how to find the roots of a polynomial equation.

 

Definition 

The roots of a polynomial are called its zeroes. It is because the roots are the x values at which the function is equal to zero. A polynomial is an expression that has two or more algebraic terms. As the name itself may suggest, poly means 'many', and 'nomial' means 'terms', hence a polynomial means it is an expression that has many terms. 

A polynomial has the following- Variable, constants and the exponents. Let's look at three examples of a polynomial to understand it in a better way. 

A – 5n + 1 (linear equation) 

B – a2 -5b + 6                

C – [d7 + (3 x 6)] – 8f

In the above examples, there are constants such as 1, 2,3,5,6, 7, and 8. The variables are a, b, d and f. The exponents are 2 and 7 as in a2 and d7. The constants can be any single or double-digit numbers; the variables can be a g, m, n or p. The same applies to the exponents – it can be one as in n, two as in m2, and so on.                                    

You can find the roots of a polynomial using several techniques. Factoring is also a method that is in use. A graph is also in use to find roots of a polynomial. Here, we shall dwell on some frequent-in-use ways. Also, it is essential to bear in mind the following: 

  1. Polynomials are terms that have only positive integer exponents. 

  2. Polynomials get the operations of addition, subtraction, and multiplication 

  3. It must be possible to write an expression without division. 


About Exponents 

An exponent is a power or the degree a number (constant) is raised to. For example, 

In a2, the exponent is 2 

In a3, the exponent is 3 

In 'a' the exponent is understood to be 1, which 'one' usually is not written.   


About the Degree 

The degree is the value of the greatest exponent in the expression in the polynomial. We are not referring to the constant here. The largest exponent will tell you about the degree. For example

In m2 + m + 4, the degree is 2 (look at the largest exponent).   

In 11m2 + 9m5, the degree is 5(look at the largest exponent) 


About the Coefficient 

The coefficient is nothing but a constant. It is the number before the variable. For example, in 3g, 3 is the coefficient. 


Roots of Polynomials 

Now you are pretty familiar with many algebraic terms. Let us discuss the roots of polynomials. The roots of a polynomial are those values of the variable that cause the polynomial to evaluate to zero. You know that polynomials are the sums and the differences of the terms that are part of the polynomial expression. So, we can say that the roots of a polynomial are the solutions for any given polynomial. 

In case of an algebraic expression, such as a polynomial, that has constants and variables, we need to find the value of the unknown variable. We can find the value of a polynomial to zero if we know the roots. 

A polynomial can be zero value, even if has constants that are greater than zero, such as 10, 25, or 46. Here, in such cases, we have to search for the value of the variables which set the value of an entire polynomial expression to a zero. These values are the 'roots' or 'zeros' of the polynomial. 


Roots of Polynomial Equation  

The formula for finding the root of a linear polynomial expression is as below 

Example: am+ b = 0 is, 

m = -b/a                                            

The formula of a quadratic equation, whose degree is 2  

Example: am2 + am + p = 0 is,  

m = [- b±√ (b2-4ap]/2


How to Find the Roots of a Polynomial

Now, let us learn how to find roots of a polynomial. Let us begin with an example, pf polynomial P(y) that has a degree 1. 

P(y) = 6y + 1 

As you know, r is the root of a polynomial P(y), if P(r) = 0. So, to determine the roots of a polynomial P(y) = 0, 

6y + 1 = 0 

y = -1/6; therefore, -1/6 is the root of polynomial P(y). So, -1/6 is a root or zero of a polynomial if it is a solution for this equation. 

Let us solve another equation to understand better. 

Example: Find P(y) = y2 -2y + 15 

                                   = (y +5) (y-3) = 0; y = -5, y = 3

This second degree polynomial has two roots or two zeroes.  


FAQ (Frequently Asked Questions)

Q1. What are Not Polynomials? 

It is easier to understand what a polynomial equation is by looking. For example, any variable raised to a (-) negative exponent is not a polynomial. If the variable is in the denominator, it is not a polynomial. If the variable is inside a radical, it is not a polynomial. So if we consider 8y-2, then it is not a polynomial, because the exponent is negative. Again, if we look at 5/x3, then it is not a polynomial, because the variable is in the denominator. The third example would be something like √57y as it is under the radical sign of square root.

Q2. What are Like Terms? 

In algebra, like terms are terms in an expression that have the same variables and powers. The coefficients need not match; they can be different. So variables and/or exponents that are similar will be called like terms. Check three variables – (2/3)xy3, -6xy3, and xy3 /5. They will all be considered like terms. The common part is xy3 in all three variables. In unlike terms, the variables are different, such as m, n, g, and k. Check three terms – 5x, 6xy2, and 3y. All are non-similar and hence are called unlike terms.