Roots of Polynomials

Roots of polynomials are the solutions for any given polynomial for which we need to find the value of the unknown variable. If we know the roots, we can evaluate the value of polynomial to zero. An expression of the form a_nxⁿ + a_n-1x^n-1 + …… + a₁x + a₀, where each variable has a constant accompanying it as its coefficient is called a polynomial of degree ‘n’ in variable x. Each variable separated with an addition or subtraction symbol in the expression is better known as the term. The degree of the polynomial is defined as the maximum power of the variable of a polynomial.

For example, a linear polynomial of the form ax + b is called a polynomial of degree 1. Similarly, quadratic polynomials and cubic polynomials have a degree of 2 and 3 respectively.

A polynomial with only one term is known as a monomial. A monomial containing only a constant term is said to be a polynomial of zero degrees. A polynomial can account to null value even if the values of the constants are greater than zero. In such cases, we look for the value of variables which set the value of entire polynomial to zero. These values of a variable are known as the roots of polynomials. Sometimes they are also termed as zeros of polynomials.

Roots of Polynomials Formula

The polynomials are the expression written in the form of:
a_nxⁿ+a_n-1x^n-1+……+a₁x+a₀

The formula for the root of linear polynomial such as ax + b is

x = -b/a

The general form of a quadratic polynomial is ax² + bx + c and if we equate this expression to zero, we get a quadratic equation, i.e. ax² + bx + c = 0.

The roots of quadratic equation, whose degree is two, such as ax² + bx + c = 0 are evaluated using the formula;

x = [-b ± √(b² – 4ac)]/2a

The formulas for higher degree polynomials are a bit complicated.

Roots of three-degree polynomial

To find the roots of the three-degree polynomial we need to factorise the given polynomial equation first so that we get a linear and quadratic equation. Then, we can easily determine the zeros of the three-degree polynomial. Let us understand with the help of an example.

Example: 2x³ − x² − 7x + 2

Divide the given polynomial by x – 2 since it is one of the factors.

2x³ − x² − 7x + 2 = (x – 2) (2x² + 3x – 1)

Now we can get the roots of the above polynomial since we have got one linear equation and one quadratic equation for which we know the formula.

Also, read:

Finding Roots of Polynomials

Let us take an example of the polynomial p(x) of degree 1 as given below:

p(x) = 5x + 1

According to the definition of roots of polynomials, ‘a’ is the root of a polynomial p(x), if
P(a) = 0.

Thus, in order to determine the roots of polynomial p(x), we have to find the value of x for which p(x) = 0. Now,

5x + 1 = 0

x = -1/5

Hence, ‘-1/5’ is the root of the polynomial p(x).

Questions and Solutions

Example 1: Check whether -2 is a root of polynomial 3x³ + 5x² + 6x + 4.

Solution: Let the given polynomial be,

p(x) = 3x³ + 5x²+ 6x + 4

Substituting x = -2,

p(-2) = 3(-2)³+ 5 (-2)² + 6(-2) + 4

p(-2) = -24 + 20 – 12 + 4 = -12

Here, p(-2) ≠ 0

Therefore, -2 is not a root of the polynomial 3x³ + 5x² + 6x + 4.

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

Solution: Given x² + 2x – 15

By splitting the middle term,

x² + 5x – 3x – 15

= x(x + 5) – 3(x + 5)

= (x – 3) (x + 5)

⇒ x = 3 or x =−5

Video Lesson

Condition for Common Roots

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Frequently Asked Questions – FAQs

Q1

What are the roots of a polynomial?

Roots of a polynomial refer to the values of a variable for which the given polynomial is equal to zero. If a is the root of the polynomial p(x), then p(a) = 0.

Q2

How many roots does a polynomial have?

The number of roots of any polynomial is depended on the degree of that polynomial. Suppose n is the degree of a polynomial p(x), then p(x) has n number of roots. For example, if n = 2, the number of roots will be 2.

Q3

How to find the roots of a polynomial?

Roots of a polynomial can be found by substituting the suitable values of a variable which equate the given polynomial to zero. The factorisation of polynomials also results in roots or zeroes of the polynomial.

Q4

How do you know if a polynomial has real roots or not?

Using Descartes’s rule of signs, we can find the number of real, positive or negative roots of a polynomial.

Q5

What is the degree of a polynomial?

The highest power (or exponent) of a variable in the polynomial is called its degree. For example, 3x^2 – 5x + 2 is a polynomial with degree 2 since the highest power of x is 2.

Also Access

NCERT Solutions for class 10 Maths Chapter 2 Polynomial

NCERT Exemplar for class 10 Maths Chapter 2 Polynomial

CBSE Notes for Class 10 Maths Chapter 2 Polynomial