# Chapter 2 – Polynomials

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

As we all know, the utility of algebra is increasing for students studying mathematics. This makes it important for us to understand the basics of algebra and the terms associated with it. Polynomials is one such concept we should be familiar with. You should familiarise yourself with the concepts and make yourself strong in solving questions involving polynomials. The word ‘polynomial’ has been derived from the Greek word ‘poly’ which means many and ‘nomial’ which means terms. Therefore, polynomials represent ‘many terms’.

Polynomials is a very common term used in mathematics. In mathematics, a polynomial is an algebraic expression consisting of three types of terms: constants, exponents, and variables. A polynomial can have several variables, exponents, and constants. These terms are always combined using mathematical operations like addition, subtraction, multiplication, and division (no division operation using a variable). Understanding the concepts of polynomials is very important if a student wants to understand algebra as a whole.

#### Notation of a Polynomial

For solving linear polynomials, you must be aware of the basic terms used to frame a polynomial.

Variables: a, b, c, x, y, z, etc.

Constants: 1, 2, 3, 4, 5, etc. (Constants can be the coefficients of a variable like 2x, 5a, 6c, etc.)

Exponents: x^6 is an example of an exponent.

Polynomial: x^2 – 5x + 6, 2z – 8

#### Degrees of a Polynomial

Let us take the example of the above polynomial.
P(x) = x^2 – 5x + 6
In the above polynomial equation, the highest exponent of a variable in a monomial is 2. Therefore, the degree of the polynomial is 2.
Thus, in a polynomial expression with one variable, the highest exponent of the monomial variable is called the degree of a polynomial.

#### Terms of a Polynomial

A polynomial, as the name suggests, can have many terms. Each term in a polynomial expression is separated from the other using a mathematical operation like ‘+’ or ‘-‘.
Let us take an example.
In the polynomial expression a^3 + 6a^2 – 8a + 9, there are four different terms: a^3, 6a^2, -8a, and 9. Each of these terms is separated by a mathematical operation.

#### Types of Polynomials

Polynomials are of three types:

1. Monomials
2. Binomials
3. Trinomials

This classification is based on the number of terms in a polynomial expression. The terms in a polynomial can be combined using addition, subtraction, multiplication, and division. We cannot divide a polynomial with a variable lest it becomes a non-polynomial. 1/x, y^(-4) are examples of non-polynomials.

Monomial

A monomial is a polynomial with a single non-zero term. Even a constant term is a monomial.
Example: x^2, 8c^3, 5y, 3, etc., are all monomials

Binomial

A binomial is a polynomial with exactly two terms. The terms in a binomial are a sum or difference of two monomials.
Example: -8c^3 -3, 5xy^2 + 13x^3y, etc.

Trinomial

A trinomial is a polynomial with exactly three terms. Trinomials are a combination of monomials, separated by either addition or subtraction.
Example: x^2 – 5x + 6, -5z^4 + 2x^3 – 6, etc.

#### Polynomial Equations

A polynomial equation is of the form an(xn). Here, ‘a’ is a coefficient, ‘x’ is a variable, and ‘n’ is the exponent.

If we expand the polynomial equation, we get:

F(x) = anxn + an-1xn-1 + an-2xn-2 + …….. + a1x +a0 = 0

Let us take an example. x^2 – 5x + 6 = 0 is a polynomial equation. Here, x^2 – 5x + 6 is the polynomial expression, which has been equated to 0.

#### Polynomial Functions

A polynomial function is an expression created with one or more terms. It is represented as:

P(x) = an(xn)

P(x) = anxn + an-1xn-1 + an-2xn-2 + …….. + a1x + a0

Here, ‘a’ is a coefficient, ‘x’ is a variable, and ‘n’ is the exponent.

Example: P(x) = x^2

P(x) = -8c^3 -3

#### Solving Polynomials

With the help of introductory algebra and factorization, we can solve polynomials. The first step in solving a polynomial equation is equating it with 0. We can solve two types of polynomial equation:

1. Linear polynomial equations
2. Quadratic polynomial equations

Solving Linear Polynomial Equations

A linear polynomial equation can be expressed in the form of ax + b = 0. Here, ‘a’ is the coefficient, ‘x’ is the variable, and ‘b’ is the constant. The degree of the equation is always 1.

To solve a linear polynomial equation, equate the polynomial to 0.

For example,

Solve for c in 2c – 4.

Solution:

2c – 4 = 0 (Equation should be made equal to 0)

2c = 4

C = 4/2

C = 2

Therefore, the solution for 2c – 4 is c = 2.

Solving Quadratic Polynomials

A quadratic polynomial equation is an equation with degree 2. It is expressed in the form ax^2 + bx + c = 0.

To solve this equation, first arrange the terms in descending order of degree, equate it to 0, and then perform polynomial factorization to get the solution.

Example:

Solve for x in x^2 + 6 – 5x

Solution:

Let us first arrange them in the decreasing order of degree

x^2 – 5x + 6

Now, we need to equate it to 0.

x^2 – 5x + 6 = 0

Now we need to perform factorization,

x^2 – 3x – 2x + 6 = 0

x(x – 3) – 2(x – 3) = 0

(x – 3)(x – 2) = 0

Therefore, x = 3 or 2.

#### Polynomial Operations

There are four primary operations that we can perform.

1. Addition
2. Subtraction
3. Multiplication
4. Division

Let us look at an example of addition of polynomials.

To add two polynomials, we can add the coefficients of the term with the same degree of variables. The addition of two polynomials always results in a polynomial of the same degree.

Example – The sum of two polynomials 5x + 3y^2 + 7 and 2y^2 – 4 – 8x, would be given as 5y^2 + 3 – 3x.

#### Conclusion

Understanding polynomials is very important as it gives you a basic understanding of how algebra works. It also forms a major portion of mathematical analysis conducted by students regularly. Therefore, understanding the basics is the first and foremost step.

There are a lot of other concepts relating to polynomials to be explored. For more content on the subject, go to the MSVgo application, where more exciting videos with explanatory animations and texts can be found.

Download the MSVgo app and get access to a repository of learning resources. It contains multiple videos which will help in understanding and brushing up your concepts on polynomials. With the help of these videos, you will understand the core concepts required to solve questions related to polynomials. It is very important to understand these underlying concepts in mathematics if you want to efficiently and easily solve problems.

#### FAQs

1. What is a polynomial equation?
A polynomial equation is of the form an(xn). Here, ‘a’ is a coefficient, ‘x’ is a variable, and ‘n’ is the exponent.
If we expand the polynomial equation, we get;
F(x) = anxn + an-1xn-1 + an-2xn-2 + …….. + a1x +a0 = 0
Let us take an example. x^2 – 5x + 6 = 0 is a polynomial equation. Here, x^2 – 5x + 6 is the polynomial expression, which has been equated to 0.
1. What are five examples of a polynomial?
Five examples of polynomials can be:
6x-2, x^2 – 5x + 6, x^3 – 1, X^4 + x^3 + x^2 + 1, and 2.
1. What is not a polynomial?
Expressions like 1/x, y^(-4) are examples of non-polynomials, which have negative exponents of variables.
1. How do you find the degree of polynomials?
In a polynomial expression with one variable, the highest exponent of the monomial variable is called the degree of a polynomial.
1. How do you classify polynomials?
Polynomials are of three types:
1. Monomials
2. Binomials
3. Trinomials

This classification is based on the number of terms in a polynomial expression.

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