Polynomial Equation and its Roots

Polynomial Equation and its Roots – Definition, Formula, Solution, Examples | How to find the Roots of Polynomials?

In this article, you will learn about the polynomial equation and its roots. Polynomial is one of the significant concepts of mathematics, and so are polynomial equations, the relation between numbers and variables are explained in a pattern. In Maths, there are a variety of equations formed with algebraic expressions. The form of algebraic equations is Polynomial Equations. You are very familiar with terms such as variables and exponents. Polynomials are the sum of the variables and exponents. You must have also heard of ‘term’. It is an individual part of the expression.

Polynomials are expressions that contain more than two or three terms. On this page, we see the definition of the roots of polynomial equations, formulas, examples, how to find the roots of a polynomial equation, example problems, and so on.

What is a Polynomial Equation?

A polynomial is an equation that has multiple terms made up of numbers and variables. It will have different exponents. The highest exponent is a degree of a polynomial. The degree will tell us how many roots can be present in a polynomial equation. For example, if the highest exponent is 3, then it means the equation has three roots.

The roots of the polynomial equation are the values of x whereas the value of y = 0. If we know the roots of the polynomial equation, we can use them to write the equation.

The examples of the polynomial equation is given below:
(i) 6x2 + 2x – 6 is a quadratic polynomial and 6x2 + 2x – 6 = 0 is its corresponding quadratic equation.
(ii) 2x3 + x2 + 5x – 3 is a cubic polynomial and 2x3 + x2 + 5x – 3 = 0 is its corresponding cubic equation.
(iii) x4 + x2 -3x + 4 is a cubic polynomial and x4 + x2 -3x + 4 = 0 is its corresponding cubic equation.
(iv) x5 + 2x4 + 2x3 + 4x2 + x + 2 is a cubic polynomial and x5 + 2x4 + 2x3 + 4x2 + x + = 0 is its corresponding equation.
If ‘α’ be an value of x for which f(x) will becomes zero, it means f(α) = 0, then α is said to be a root of the equation f(x), n= 0. In other words, the α is called a root of the polynomial equation then f(x) = 0 if f(α) = 0.

Roots of Polynomials

The roots of a polynomial are whose values of the variable will cause the polynomial to evaluate as zero. You know the polynomials are the sums and the differences of the terms which are part of the polynomial expression. So, the roots of a polynomial are the solutions for any given polynomial.

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

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

Roots of Polynomial Equation

The three important theorems which are relating to the roots of a Polynomial equation:

  1. A polynomial of nth degree will be factored into n linear factors.
  2.  A polynomial equation of degree is ‘n’, which means it has exactly ‘n’ roots.
  3.  If (x−r) is a factor of a polynomial, then x=r is the root of the associated polynomial equation.

The polynomials are the expression written in the form of anxn+an-1xn-1+……+a1x+a0. The formula for finding the roots of a linear polynomial expres is  below,
Example: ax+ b = 0 then x = -b/a.

The formula of the quadratic equation, whose degree is 2 . Then the general form of a quadratic polynomial is ax+ bx + c and if we equate this expression to zero, we get a quadratic equation, i.e. ax+ bx + c = 0, such as ax+ bx + c = 0 are evaluated using the formula;
Example: ax2 + bx + c = 0 is, then m = [- b±√ (b2-4ac]/2.

The formulas for the highest degree polynomials are a bit complicated.

How do you find the Roots of a Polynomial?

You will find the roots of a polynomial using several techniques. One of the methods we will be using is the Factoring method. A graph is also used to find the roots of a polynomial. Here, we shall discuss 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.

Exponents: An exponent is a power or the degree to which a number (constant) is raised. For example, a2 has an exponent is 2, and a3 has an exponent is 3.
In ‘a’ the exponent is understood to be 1, which the ‘one’ usually is not written.

Degree: The degree is the value of the greatest or highest exponent of the expression in the polynomial. Here, we are not referring to the constant. About the degree, the largest exponent will explain. For example m2 + m + 4, the degree is 2 (look at the largest exponent).

Coefficient: The coefficient is nothing but a constant. It is the number before the variable. For example, in 3g, the coefficient value will be 3.
Now, see how to find the roots of a polynomial. Let us starts with an example, the polynomial P(y) has a degree of 1 that is P(y) = 6y + 1.
As we know, r is the root of a polynomial P(y), if P(r) = 0. So, it can determine the roots of a polynomial P(y) = 0, i.e., 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.

Examples: (i) Let f(x) = 4x3+12x2-4x -12. As the equation is 4(1)3 + 12(1)2 – 4(1) – 12 = 4 +12- 4-12= 0, that is  f(1) = 0, f(x) = 0. So, the roots of x is 1.
(ii) Let f(x) = x2 – 2x-3. After substituting the equation is (-1)2-2(-1)-3 = 1 + 2 – 3 = 0, that is f(-1) = 0, f(x) = 0 it has a root of x = -1.
(iii) Let f(x) = x4 + x3 – 2x2 + 4x – 24. As (2)4 +(2)3-2(2)2 +  4(2) – 24 = 16 + 8 – 8+8 + 8 = 0, that is f(2) =0, f(x) then it has a root x = 2.
(iv) Let f(x) = x3+x2-x-1. After substituting, the equation is (1)3+(1)2 – (1) – 1 = 1+1 – 1 – 1 = 0, that is f(1) = 0, f(x) = 0. So, the root of x is 1.

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Examples of How to Solve Polynomial Equations

Problem 1:
Maria came across a problem according to which (2x+6) and (x−8) are the roots of the polynomial equation. Find the polynomial equation when its respective roots are given?

As given in the question,
The Roots of equations are (2x+6) and (x−8).
Now, we need to find the polynomial equation using the given roots.
For any polynomial of P(x), the polynomial equation will be P(x) = 0
This mean (2x+6) (x-8) = 0 will give the polynomial equation. Let’s solve it.
2x(x−8) +6(x−8) = 0
2x3−16x+6x−48= 0
Therefore, the polynomial equation of the given roots is 2x3−10x−48=0.

Problem 2:
Find the roots of the polynomial x2 + 2x – 15.

In the given question, the equation is x2 + 2x – 15.
Now, we need to find the roots of the polynomial.
First, we can split the middle term, then we get
x2 + 5x – 3x – 15
x(x + 5) – 3(x + 5)
(x – 3) (x + 5)
The value of x is 3 or x is −5.
Thus, the roots of the polynomial are 3, -5.

FAQs on Polynomial Equation and its Roots

1. What is the Difference between Polynomial and Equation?
A polynomial is the parent term used to describe a certain type of algebraic expression that contain variables, and constants, which involves the operations of addition, subtraction, multiplication, and division, along with only positive powers associated with the variables. An equation will be a mathematical statement that has an ‘equal to’ sign between the two algebraic expressions which have equal values.

2. How do you find the roots of a polynomial?
The Roots of a polynomial can be found by substituting the suitable values of a variable that equates the given polynomial to zero. The factorization of polynomials also results in roots or zeroes of the polynomial.

3. How do you know if a polynomial has real roots or not?
By using Descartes’s rule of signs, we can find the number of real, positive, or negative roots of a polynomial.

4. What is the degree of a polynomial?
The highest power (or exponent) of a variable in the polynomial is called its degree. Consider an example, 3x2 – 5x + 2 is a polynomial it has a degree of 2. Hence, the highest power of x is 2.

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