class 9

 CONTENTS OF THIS PAGE:

Algebraic Identities (5 to 7)

Exponents

Laws of exponents

How to simplify exponents?

What is a rational number?

What is an irrational number?

How to express decimal expansion in the form p/q?

How to rationalise the denominator?

How to simplify irrational numbers?

How to simplify rational powers?

What is a polynomial?

Polynomials

How to find the roots or zeros of a polynomial?

Remainder Theorem

Factor Theorem

EXPONENTS:  

It is form of expressing numbers in terms of powers.  For example the number 8 is 2 x 2 x 2 that is 2 multiplied 3 times.  Here when 8 is written in the exponential form, we say 2 is the base and 3 is the exponent (power).

NUMBERS WRITTEN IN EXPONENTIAL FORM

8 = 2x2x2 = 23

81 = 3x3x3x3 = 34

216 = 2x2x2x3x3x3

     = 23 x 33

RATIONAL NUMBERS:

Numbers which can be written in the form p/q (where p and q are integers and q not equal to zero) are called rational numbers. A number that can be expressed as a ratio (quotient) of two integers is a rational number. Examples: 3/4, -6/7 

The decimal expansion of rational numbers are of two types:

1. Terminating decimal

For example, the decimal expansion of 1/8 is 0.125

The decimal expansion of 125/4 is 31.25

2. Non terminating repeating decimal

For example, the decimal expansion of 2/3 is 0.666666..... 

The decimal expansion of 43/99 is 0.43434343....

IRRATIONAL NUMBERS: 

Numbers that cannot be written in the form p/q are called irrational numbers.  In other words, a number that cannot be expressed as a ratio (quotient) of two integers is an irrational number.  The decimal expansion of irrational numbers will be non terminating non repeating.  

POLYNOMIALS:

A polynomial is an expression which consists of variables, constants and exponents associated with the basic operations of Mathematics like addition, subtraction, multiplication and division.  In other words, an expression containing many terms is called a polynomial.  Examples: 5x+2, 3x-1. 

A polynomial is denoted by p(x). The value of x for which the polynomial p(x) becomes 0 is called the root or zero of the polynomial p(x).  Let us see some examples:

REMAINDER THEOREM: 

It states that when a polynomial p(x) is divided by x-a, where a is any real number, then p(a) is the remainder of p(x). An example is given below:

FACTOR THEOREM:

If a is a zero of a polynomial p(x), then x-a is a factor of p(x).  In other words, if p(a) = 0, then x-a is a factor of p(x).