Table of contents  
Polynomials  
Types of Polynomials based on Degree  
Types of Polynomials based on the Number of Terms  
Polynomial Operations  
Solved Examples 
Any expression that contains constants, variables as well as exponents which can be either added, subtracted, multiplied or divided.
But there is a catch, it must follow these rules:
1. Cannot have an infinite number of terms
2. A variable’s exponent can only be a whole number i.e., 0,1,2,3…etc.
3. Cannot be divisible by a variable
Polynomial
If p(x) is a polynomial in x, the highest power of x in p(x) is called the degree of polynomial p(x).
Examples:
1. 3x^{4}+2x^{2}+6x = 1, the highest power of the given polynomial equation is 4. Therefore, its degree is 4.
2. x^{5}y^{3}z + 2xy^{3}+4x^{2}yz^{2}
because this equation contains several variables ( that is, x,y, and z) we need to add up the degrees of the variables in each of the terms in order to find the degree of the variable. Please note, it does not matter if they are differing variables.
as a result, a 1^{st} term (5+3+1), the 2^{nd} term (1+3) and the 3^{rd} term (2+1+2).
The highest total is 9, of the 1st term, consequently, its degree is 9.
Here's a video about Polynomials:
1. Linear Polynomial: Polynomial equation with degree 1 is called a linear polynomial.
Example: x – 3 = 0
2. Quadratic Polynomial: Polynomial equation with degree 2 is called a quadratic polynomial.
Example: x^{2}+2x+7 = 0.
3.Cubic Polynomial: Polynomial equation with degree 3 is called a cubic polynomial.
Example: 3x^{3}–5x^{2}+7x+5 = 0.
1. Monomial: A polynomial with just one term.
Example: 2x, 6x^{2}, 9xy
2. Binomial: A polynomial with two terms.
Example: 4x^{2}+x, 5x+4
3. Trinomial: A polynomial with three terms.
Example: x^{2}+3x+4
There are four main polynomial operations which are:
Each of the operations on polynomials is explained below using solved examples.
To add polynomials, always add the like terms, i.e. the terms having the same variable and power. The addition of polynomials always results in a polynomial of the same degree. For example,
Example: Find the sum of two polynomials: 5x^{3 }+ 3x^{2}y + 4xy − 6y^{2 }and 3x^{2 }+ 7x^{2}y − 2xy + 4xy^{2 }− 5
Solution:
First, combine the like terms while leaving the unlike terms as they are. Hence,
(5x^{3 }+ 3x^{2}y + 4xy − 6y^{2}) + (3x^{2 }+ 7x^{2}y − 2xy +4xy^{2 }−5)
= 5x^{3 }+ 3x^{2 }+ (3+7)x^{2}y + (4−2)xy + 4xy^{2 }− 6y^{2 }− 5
= 5x^{3 }+ 3x^{2 }+ 10x^{2}y + 2xy + 4xy^{2 }− 6y^{2 }− 5
Subtracting polynomials is similar to addition, the only difference being the type of operation. So, subtract the like terms to obtain the solution. It should be noted that subtraction of polynomials also results in a polynomial of the same degree.
Example: Find the difference of two polynomials: 5x^{3 }+ 3x^{2}y + 4xy − 6y^{2}, 3x^{2 }+ 7x^{2}y − 2xy + 4xy^{2 }− 5
Solution:
First, combine the like terms while leaving the unlike terms as they are. Hence,
(5x^{3 }+ 3x^{2}y + 4xy −6y^{2})  (3x^{2 }+ 7x^{2}y − 2xy + 4xy^{2 }− 5)
= 5x^{3}3x^{2 }+ (37)x^{2}y + (4+2)xy 4xy^{2 }− 6y^{2 }+ 5
= 5x^{3 } 3x^{2 } 4x^{2}y + 6xy  4xy^{2 }− 6y^{2 }+ 5
Multiplication of Polynomials
Two or more polynomial when multiplied always result in a polynomial of higher degree (unless one of them is a constant polynomial). An example of multiplying polynomials is given below:
Example: Solve (6x − 3y)x(2x + 5y)
Solution:
⇒ 6x ×(2x + 5y)–3y × (2x + 5y) ——— Using distributive law of multiplication
⇒ (12x^{2 }+ 30xy) – (6yx + 15y^{2}) ——— Using distributive law of multiplication
⇒12x^{2 }+ 30xy – 6xy – 15y^{2 }—————– as xy = yx
Thus, (6x−3y)×(2x+5y)=12x^{2}+24xy−15y^{2}
Division of two polynomial may or may not result in a polynomial. Let us study below the division of polynomials in details. To divide polynomials, follow the given steps:
Polynomial Division Steps:
If a polynomial has more than one term, we use the long division method for the same. Following are the steps for it.
Example:
Given two polynomial 7s^{3}+2s^{2}+3s+9 and 5s^{2}+2s+1.
Solve these using mathematical operation.
Solution:
Given polynomial:
7s^{3}+2s^{2}+3s+9 and 5s^{2}+2s+1
Polynomial Addition: (7s^{3}+2s^{2}+3s+9) + (5s^{2}+2s+1)
= 7s^{3}+(2s^{2}+5s^{2})+(3s+2s)+(9+1)
= 7s^{3}+7s^{2}+5s+10
Hence, addition result in a polynomial.
Polynomial Subtraction: (7s^{3}+2s^{2}+3s+9) – (5s^{2}+2s+1)
= 7s^{3}+(2s^{2}5s^{2})+(3s2s)+(91)
= 7s^{3}3s^{2}+s+8
Hence addition result in a polynomial.
Polynomial Multiplication:(7s^{3}+2s^{2}+3s+9) × (5s^{2}+2s+1)
= 7s^{3} (5s^{2}+2s+1)+2s^{2} (5s^{2}+2s+1)+3s (5s^{2}+2s+1)+9 (5s^{2}+2s+1))
= (35s^{5}+14s^{4}+7s^{3})+ (10s^{4}+4s^{3}+2s^{2})+ (15s^{3}+6s^{2}+3s)+(45s^{2}+18s+9)
= 35s^{5}+(14s^{4}+10s^{4})+(7s^{3}+4s^{3}+15s^{3})+ (2s^{2}+6s^{2}+45s^{2})+ (3s+18s)+9
= 35s^{5}+24s^{4}+26s^{3}+ 53s^{2}+ 21s +9
Polynomial Division: (7s^{3}+2s^{2}+3s+9) ÷ (5s^{2}+2s+1)
(7s^{3}+2s^{2}+3s+9)/(5s^{2}+2s+1)
This cannot be simplified. Therefore, the division of these polynomials does not result in a polynomial.
Other topics covered under Polynomials:
Geometrical Meaning of the Zeroes of a Polynomial
13 videos79 docs29 tests

1. What are the different types of polynomials based on their degree? 
2. How are polynomials classified based on the number of terms they have? 
3. What are some common operations performed on polynomials? 
4. Can you provide an example of a linear polynomial? 
5. How can polynomials be useful in reallife applications? 
13 videos79 docs29 tests


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