Polynomials From Basics! Types, Formula, Rules, Operation with Solved Examples

Polynomials can be defined as algebraic expressions that contain variables and constants. Variables are also sometimes called indeterminates.

What are Polynomials?

Polynomials are like the heart of mathematics. They are basically used to express numbers in the field of mathematics and are referred to as very important in some branches of mathematics, such as calculus and algebra.

Polynomial is made up of two words:-
Poly = “many”
Nominal = “terms.”

The terms are combined using mathematical operations such as addition, subtraction, multiplication, and division.
For example = 32x + 2 and 3x + 5y are Polynomials.
1. 32x + 2 = In this example, there is one variable x and two terms 32x and 12.
2. Similarly, 3x + 5y. In this example, there are two variables x and y.It contains two terms 3x and 5y.

The Standard form of Polynomial

Definition 1 = The standard form of a polynomial basically deals with how to write a polynomial in the descending power of the variable.

Definition 2 = The Standard form for writing a polynomial is to write the terms with the highest degree first.

Let us understand with an example:-

1. Convert the polynomial 8 + 2x ³ + x ²  in the standard form.

Step 1 = First, check the degree of the polynomial. In the given polynomial, the degree is 3.

Step 2 = Write the term with degree 3.

Step 3 = Now, check if there is a term with an exponent of variable less than 3 which is 2 and 1, and note it down next.

Step 4 = At last, write the term with the exponent of the variable as 0, which is the constant term.

Step 5 = 2x ³ + x  ²  + 8 is the standard form of the polynomial.

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Degree of a Polynomial

Definition = The highest exponent of the variable in a polynomial can be defined as the degree of a polynomial.
(Exponent is a method of expressing large numbers in terms of powers, for example, when 3 is multiplied by itself 4 times, which is 3 × 3 × 3 × 3. This can be written as 34.)

The degree is used to tell the maximum number of solutions a polynomial equation can have.
Degree = 3 means the equation can have 3 solutions.

Let us understand with the help of some examples:-

1. A polynomial 3x ⁵  + 7 has a degree equal to five.
2. 2x ² + 8 has a degree equal to two.

Now, let us calculate the degree of a polynomial having more than one variable, simply add the powers of all the variables in a term. So, we will get the final degree of the given polynomial p.

The degree of the polynomial 2x ² y ³ + 7x ⁴ y is 10.

Read More : METHODS FOR SOLVING QUADRATIC EQUATION

Types of Polynomials

Polynomials can be divided on the basis of their degree and power.

1. Based on the Number of Terms.
2. Based on Number of Degrees.

A. Polynomials Types: Base on the Number of Terms

According to the number of terms, polynomials can be categorized into three types:-

1. Monomials
2. Binomials
3. Trinomials

1. Monomial
A Monomial can be defined as an expression containing a single term. ( Note – The single term must be non-zero. ) For example = x, 5xy, 4y² and 6y³

2. Binomial
A Binomial can be defined as an expression containing two terms. For example =  Y+5, x³+9 and  3x⁴-2

3. Trinomial
A Trinomial can be defined as an expression that contains three terms. For example =

3x⁴ + 8x – 5, x + y + 2z and 3x + y – 5.

Read More : Sum of cubes of n natural numbers

Polynomials Types: Based on the Number of Degrees

According to the number of degrees, polynomials can be categorized into four major types:-

1. Zero Polynomial or Constant polynomial
2. Linear polynomial
3. Quadratic polynomial
4. Cubic polynomial

1. Zero Polynomial

A zero polynomial can be defined as a polynomial whose degree is equal to zero. All the constants are examples of zero polynomials. For example = 3, 6 and 9.

2. Linear Polynomial

Linear polynomials can be defined as polynomials whose degree is equal to one. For example = x – 4.

3. Quadratic Polynomial

Quadratic polynomials can be defined as polynomials whose degree is equal to two.For example = 2q² – 7.

4. Cubic polynomials

Cubic polynomials can be defined as polynomial whose degree is equal to three.For example = 7x³ – y – 4

Read More : Fractions From Basics. Understand What is Fraction, Types with Examples

Operations on Polynomials

Any basic operation that can be performed on numbers can be performed on the polynomials too. So, there are four types of operations that can be performed on polynomials:-

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

1. Addition of Polynomials

It is one of the basic algebraic operations which is used to increase and decrease the value of the polynomials.
The rule to add numbers and polynomials is exactly the same.

To add the polynomials, you should simply add on the like terms (like terms means the terms having the same variable and same power).
For eg –

a. Add 2x² + 4x + 5 and 3x²

= 5x² + 4x + 5

b. Add 4x³+ x² + 8 and 2x³ + 4x² + 4x

= 6x³ + 5x² + 4x + 8

Read More : Subtraction of Fractions

Subtraction of Polynomials

It is also one of the basic algebraic operations which are used to increase and decrease the value of the polynomials.
The rules to subtract numbers and polynomials are also similar.

To subtract the polynomials, you should simply subtract the like terms (like terms mean the terms having the same variable and same power).
For eg –

a. Subtract 4x² + 4x + 5 and 3x²

= x² + 4x + 5

b. Subtract  4x³+ x² + 8 and 2x³ + 4x² + 4x

= 2x³ – 3x² + 4x + 8

Read More : Multiplication of Fractions

3. Multiplication of Polynomials

In the multiplication of polynomials, two or more two polynomials are multiplied and always give a result with a higher degree.

In polynomials, you can simply multiply the like and unlike terms both together. (like terms mean the terms having the same variable and same power and vice versa…).

For eg –

a. Multiply 2x² and 3x²

= 6x⁴

b. Multiply 4x³ and 2x³ + 4x² + 4x

Step 1 = firstly multiply 4x³ with 2x³= 8x⁶  (in multiplication powers add on)

Step 2 = then, multiply 4x³ with 4x² = 16x⁵

Step 3 = At last, multiply  4x³ with 4x = 16x⁴

Now, combine the terms Answer is 8x⁶ + 16x⁵ + 16x⁴.

Read More : Multiplication of Fractions

4. Division of Polynomials

When we are dividing the polynomials, two or more than two polynomials can be divided and always give a result with a smaller degree.

In Polynomials you can simply divide the like and unlike terms both together. (like terms mean the terms having the same variable and same power and vice versa…).
For eg –

a. Divide 2x² and x

= 2x

In this blog, we have studied the basic concepts of Polynomials, their types, degrees, and operations on polynomial expressions. We will cover the simplification of polynomials in the next session.

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Practice Question:

1. Add:  2x²+3x and 4x²

2. Add: 4x²+2x+7 and 3x²+3x+7

3. Subtract: 5x²+3x from 7x²

4. Multiply: 4x²(3x²+5x)

5. Divide:   8x²/2x

Quizzes for you:-

Question 1: Which of the following is an example of a monomial?

1. 2x²
2. 3x  ( Correct Answer )
3. 4x³

Question 2: In broadly, how many types of polynomials based on the degree are there?

Options

1. 4 ( correct answer )
2. 5
3. 2
4. 1

Question 3: What is the sum of 3x²+5x and 2x²+6x?

Options

1. 4x²+6x
2. x²-x
3. 5x²+11x ( Correct answer )
4. -x²+11x