#### 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 43

4 ∗ 4 is written as 42

a ∗ a ∗ a is written as a3

In a3, 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 x2 y3 is also a monomial.

#### Polynomial

A polynomial is an indicated sum of monomials.

Examples:

• 3x + 7

• (4/7)x2 + 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.

1.    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.
2.    A + B = B + A (Commutative Property)
3.    (A + B) + C = A + (B + C) (Associative Property)
4.    The zero polynomial is the identity element under addition.
5.    If 'A' is a polynomial, its additive inverse is -A. Thus, every polynomial has an additive inverse.

Example:

If A = 3x3 + 4x2 - x - 1

B = 4x3 -3 x2+ 4x + 5

Find A + B and B + A

A + B = (3x3 + 4x2 - x - 1) + (4 x3 - 3 x2+ 4x + 5)

= (3 + 4) x3 + (4 - 3) x2 + (- 1 + 4) x + (- 1 + 5)

= 7x3 + x2+ 3x + 4

B + A = ( 4x3 - 3x2 + 4x + 5) + (3x3 + 4x2 - x - 1)

= (4x3 + 3x3) + (- 3x2 + 4x2) + (4x - x) + 5 - 1

= 7x3 + x2 + 3x + 4

It can be seen that A + B = B + A.

#### Try these questions

State the coefficients and degrees of the polynomials.

1.    10x5

2.    Answer: Coefficient = 10; degree = 5

3.    - 2.51 x4

4. Answer: Coefficient = -2.51; degree = 4

5. – 8

6. Answer: Coefficient = - 8; degree = 0

7. √3x2

8. Answer: Coefficient = √3; degree = 2

Find the value of monomial when x = 3, 4.

9. 2x2

10. 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

11. 3x2

12. Answer: When x = 2,
the value of
3x2 = 3 ∗(2)2 = 12

When x = 3;
the value of 3x2
= 3 ∗(3)2= 27

When x = - 1.5,
the value of 3x2
= 3 ∗(-1.5)2 = 6.75

13. -1.2 x2

14. Answer: When x = 2,
the value of -1.2 x2
= -1.2 ∗(2)2 = -4.8

When x = 3,
the value of -1.2 x2
= -1.2 ∗(3)2= - 10.8

When x = - 1. 5
the value of -1. 2x2
= - 1.2 ∗( - 1.5 )2 = - 2.7

15. 1/2x3

16. Answer: When x = 2,
the value of 1/2 x3
= 1/2 ∗2 ∗2∗2 = 4

When x = 3 ;
the value of 1/2 x3
= 1/2 ∗3∗ 3 ∗3 = 13.5

When x = -1.5 the value of 1/2 x3
= 1/2 ∗-1.5 ∗-1.5 ∗-1.5
= -1.6875

17. 2x3

18. Answer: When x = 2,
then the value of 2x3
= 2 ∗(2)3 = 2∗ 8 = 16

When x = 3,
then the value of 2x3
= 2 ∗(3)3 = 2 ∗27 = 54

When x = - 1.5,
then the value of 2x3
= 2 ∗(-1.5)3 = - 6.75

Simplify

19. - 3x2 + ( 6x2 ) - ( -0.5x2 ) + ( 1.5x2)

20. Answer: = - 3x2 + 6x2 + 0.5x2 + 1.5x2
= ( - 3 + 6 + 0.5 + 1.5 ) x2 = 5x2

21. ( - 3x ) + ( - 4x ) - ( 4.5 ) x + ( 2.5x )

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

23. ( 3x ) + ( - 4x ) - ( - 3x ) + ( - 7x )

24. Answer: = 3x - 4x + 3x - 7x
= (3 - 4 + 3 - 7) x = - 5x

25. ( - 5x2 ) + ( 5.2x2 ) + ( 1.5x2 ) - ( 0.7x2 )

26. Answer: = ( - 5 + 5.2 + 1.5 - 0.7 ) x2
= ( 6.7 - 5.7 ) x2 = (1) x2 = x2