The document presents an educational overview of polynomials, defining them as expressions with variables and coefficients that involve basic arithmetic operations but are not divided by a variable. It categorizes polynomials into three types: monomials, binomials, and trinomials, and explains their properties, degrees, and methods for solving linear and quadratic equations. Examples and step-by-step solutions illustrate the concepts, aiming to enhance understanding of polynomial equations.

1. Presentation made by Alankrit Wadhwa
Class IX

2. In Mathematics, a polynomial is an expression consisting of variables and
coefficients which involve only the operations of addition, subtraction,
multiplication and division but is never divided by a variable. We use
polynomials in areas/fields of mathematics and science. They are used to
form various equations which are capable of solving problems from
elementary word problems to complicated and confusing scientific
problems.
A Polynomial is made up of two terms, namely Poly(meaning “many”) and
nomial(meaning “terms”).
Here are a few examples for polynomials:-
7a3+ 2a2+6a+8, 5a2+2a+1 etc.

3. Polynomials are of three types and are classified based on the number of the terms in it. The three types of
polynomials are:-
 Monomial
 Binomial
 Trinomial
These polynomials can be combined using addition, subtraction, multiplication and division but is never divided by
a variable.
• Monomial
A monomial is an expression which contains only one term. For an expression to be a monomial, the single term
should be a non-zero term. Some examples of monomials are: 5x, 7, 6z2, -7xy, 9xyz etc.
• Binomial
A binomial is a polynomial expression which contains exactly and only two terms. A binomial can be considered as
a sum or difference between two or more monomials. Some examples of binomials: -3x+3, 78x2+19x, xy2+2y etc.
• Trinomial
A trinomial is an expression which is composed of three terms. Some examples of trinomials: -2x3+39x+4z,
x2+2x+10, 4y2+8x+10 etc.

4. Consider a polynomial p(x) = 5x3-2x3+3x-2
If we replace x by 1 everywhere in p(x), we get
p(1) = 5x(1)3-2x(1)2+3x(1)-2
= 5-2+3-2
= 4
So, we say that the value of p(x) at x=1 is 4.
Similarly,
p(0) = 5(0)3-2(0)2+3(0)-2
= -2

5. The degree of a polynomial is defined as the
highest degree of a monomial within a polynomial.
Thus, a polynomial equation having one variable
which has the largest exponent is called a degree
of the polynomial. Refer to the table below to
understand it better.
Polynomial Degree Example
Constant or Zero 0 6, 10, 100
Linear Polynomial 1 3x+1, 7z+6, 8y+4
Quadratic Polynomial 2 4x2+1x+1, 7y2+2y+3
Cubic Polynomial 3 6x3+3x2+9x+1
Quartic Polynomial 4 6x4+3x3+3x2+2x+1

6. Observe the image above properly and try to understand polynomials, in your own words.
Here’s a simple definition which you can remember easily once you’ve seen the image above.
A polynomial is an expression which consists of variables, constants, exponents and coefficients.

7. Any polynomial can be easily solved using basic algebra
and factorization concepts. While solving
the polynomial equation, the first step for us is to take
the RHS(Right-Hand Side) as 0.
The explanation of a polynomial solution is explained in
two different ways:
 Solving Linear Polynomials
 Solving Quadratic Polynomials

8.  Solving Linear Polynomials
Solving linear polynomials and getting the solution is really easy and simple.
Firstly, isolate the variable term and make the equation as equal to zero.
Then solve it as basic algebra operation. Let us take an example and apply
these same steps to get an answer for the equation.
Example:- Solve 3x-9
Solution:- Firstly, make the equation as 0,
LHS = RHS
3x-9 = 0
3x = +9
x = 9/3
x = 3
We got x=3 as the solution for the equation.
We can solve all of the linear equations this easily by applying this simple method.

9.  Solving Quadratic Polynomials
To solve a quadratic polynomial, first, rewrite the expression in the descending order
of degree.
Then, equate the equation and perform polynomial factorization to get the solution of
the equation.
Let us understand this by the help of an example.
Example:- Solve 3x2-6x+x3-18
Solution:- x3+3x2-6x-18 = 0
x2(x+3)-6(x+3) = 0
(x2-6)(x+3)
Solution will be x = -3 then,
x2 = 6
x = √6

10. Hope you understood the chapter well…