Definitions
Let's learn a few definitions.
Factor of the number
When two or more numbers are multiplied, each of the numbers is called a factor of the number.
For example, in the product 5 ∗ 11 = 55, 5 and 11 are factors of 55.
Coefficient
Each factor is the coefficient of the product of other factors.
For example, in a term 3xy

3 is a coefficient of xy

x is a coefficient of 3y

y is a coefficient of 3x

xy is a coefficient of 3
Generally, the numerical part of a term is called the numerical term of its coefficient.
Thus in the term 3xy, 3 is the numerical coefficient.
Exponent
Sometimes, the products are written as powers.
For example, 4 ∗ 4∗ 4 is written as 4^{3}
4 ∗ 4 is written as 4^{2}
a ∗ a ∗ a is written as a^{3}
In a^{3}, 3 is called the exponent or power and 'a' is called the base; the exponent 3 dictates the number of times the
base 'a' occurs as a factor in the product.
Monomial
A monomial is a term that is either a number or a variable with positive integral index or an indicated product of a
number and one or more variables.
Examples:

7 is a monomial since it is a number.

p is a monomial since it is a variable.

7p is a monomial since it is an indicated product of a number 7 and p.

7pq is also a monomial since it is a product of 7 and the variable 'pq'. 3/4 x^{2} y^{3} is also a monomial.
Polynomial
A polynomial is an indicated sum of monomials.
Examples:
3x + 7
(4/7)x^{2} + 6x  8
Degree of polynomials
The degree of a polynomial is the greatest degree of its various terms.
Example:
2x + 3 has two terms, namely 2x and 3. The degree of 2x is 1. The degree of 3 is 0. The greatest of the two
degrees is 1.
Addition properties of polynomials
 A and B are two polynomials. By adding them, we get (A+B), which is also a polynomial. Hence, the set of polynomials has the Closure Property.
 A + B = B + A (Commutative Property)
 (A + B) + C = A + (B + C) (Associative Property)
 The zero polynomial is the identity element under addition.
 If 'A' is a polynomial, its additive inverse is A. Thus, every polynomial has an additive inverse.
Example:
If A = 3x^{3} + 4x^{2}  x  1
B = 4x^{3} 3 x^{2}+ 4x + 5
Find A + B and B + A
A + B = (3x^{3} + 4x^{2}  x  1) + (4 x^{3}  3 x^{2}+ 4x + 5)
= (3 + 4) x^{3} + (4  3) x^{2} + ( 1 + 4) x + ( 1 + 5)
= 7x^{3} + x^{2}+ 3x + 4
B + A = ( 4x^{3}  3x^{2} + 4x + 5) + (3x^{3} + 4x^{2}  x  1)
= (4x^{3} + 3x^{3}) + ( 3x^{2} + 4x^{2}) + (4x  x) + 5  1
= 7x^{3} + x^{2} + 3x + 4
It can be seen that A + B = B + A.
Try these questions
State the coefficients and degrees of the polynomials.

10x^{5}
Answer: Coefficient = 10; degree = 5

 2.51 x^{4}
Answer: Coefficient = 2.51; degree = 4

– 8
Answer: Coefficient =  8; degree = 0

√3x^{2}
Answer: Coefficient = √3; degree = 2
Find the value of monomial when x = 3, 4.

2x^{2}
Answer: when x = 3
2 ∗(3)^{2} = 2 ∗9 = 18
when x = 4
2 ∗(4)^{2} = 2 ∗16 = 32
Find the values of the monomials when x = 2, 3,  1.5

3x^{2}
Answer: When x = 2,
the value of
3x^{2} = 3 ∗(2)2 = 12
When x = 3;
the value of 3x^{2}
= 3 ∗(3)^{2}= 27
When x =  1.5,
the value of 3x^{2}
= 3 ∗(1.5)^{2} = 6.75

1.2 x^{2}
Answer: When x = 2,
the value of 1.2 x^{2}
= 1.2 ∗(2)^{2} = 4.8
When x = 3,
the value of 1.2 x^{2}
= 1.2 ∗(3)^{2}=  10.8
When x =  1. 5
the value of 1. 2x^{2}
=  1.2 ∗(  1.5 )^{2} =  2.7

1/2x^{3}
Answer: When x = 2,
the value of 1/2 x^{3}
= 1/2 ∗2 ∗2∗2 = 4
When x = 3 ;
the value of 1/2 x^{3}
= 1/2 ∗3∗ 3 ∗3 = 13.5
When x = 1.5 the value of 1/2 x^{3}
= 1/2 ∗1.5 ∗1.5 ∗1.5
= 1.6875

2x^{3}
Answer: When x = 2,
then the value of 2x^{3}
= 2 ∗(2)3 = 2∗ 8 = 16
When x = 3,
then the value of 2x^{3}
= 2 ∗(3)3 = 2 ∗27 = 54
When x =  1.5,
then the value of 2x^{3}
= 2 ∗(1.5)3 =  6.75
Simplify

 3x^{2} + ( 6x^{2} )  ( 0.5x^{2} ) + ( 1.5x^{2})
Answer: =  3x^{2} + 6x^{2} + 0.5x^{2} + 1.5x^{2}
= (  3 + 6 + 0.5 + 1.5 ) x^{2} = 5x^{2}

(  3x ) + (  4x )  ( 4.5 ) x + ( 2.5x )
Answer: = (  3  4  4.5 + 2.5 ) x =  9x

( 3x ) + (  4x )  (  3x ) + (  7x )
Answer: = 3x  4x + 3x  7x
= (3  4 + 3  7) x =  5x

(  5x^{2} ) + ( 5.2x^{2} ) + ( 1.5x^{2} )  ( 0.7x^{2} )
Answer: = (  5 + 5.2 + 1.5  0.7 ) x^{2}
= ( 6.7  5.7 ) x^{2} = (1) x^{2} = x^{2}