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  Rational Numbers Definition Rational    numbers are one very common type of number that we usually study after integers in math. These numbers are in the form of p/q, where p and q can be any integer and q ≠ 0. Types of Rational Numbers There are different types of rational numbers. We shouldn't assume that only fractions with integers are rational numbers. The different  types of rational numbers are: Integers like -2, 0, 3 etc. Fractions whose numerators and denominators are integers like 3/7, -6/5, etc.      T erminating decimals like 0.35, 0.7116, 0.9768, e Non-terminating decimals with some repeating patterns (after the decimal point) such as 0.333..., 0.141414..., etc. These are popularly known as non-terminating repeating decimals Smallest Rational Number Look at the chart given below to understand the difference between rational numbers and irrational numbers along with other types of numbers pictorially. Standard Form of Rational Numbers The...

  Irrational Numbers Irrational numbers are those real numbers that cannot be represented in the form of a ratio.  In other words, those real numbers that are not rational numbers are known as irrational numbers Irrational Numbers Definition An irrational number is a real number that cannot be expressed as a ratio of integers. for example, √ 2 is an irrational number. Again, the decimal expansion of an irrational number is neither terminating nor recurring. In other words, we can say that irrational numbers cannot be represented as the ratio of two integers. Are Irrational Numbers Real Numbers? In Mathematics, all the irrational numbers are considered as real numbers, which should not be rational numbers.  It means that irrational numbers cannot be expressed as the ratio of two numbers.  The irrational numbers can be expressed in the form of non-terminating fractions and in different ways.  For example, the square roots which are not perfect squares will always ...

  FACTORIZATION - USING IDENTITY  An identity is an equality that remains true regardless of the values chosen for its variables. ALGEBRAIC IDENTITIES  EXAMPLE 1  Factorize:   9X 2  + 12XY + 4Y 2  [  a 2  + 2ab + b 2  = (a + b) 2  ]    9X 2  + 12XY + 4Y 2  = (3x) 2  + 2(3x)(2y) + (2y) 2   9X 2  + 12XY + 4Y 2  = (3x + 2y) 2  EXAMPLE 2  Factorize:  25a 2  – 10a + 1 [ a 2  – 2ab + b 2  = ( a – b) 2  ] 25a 2  – 10a + 1 = (5a) 2  – 2(5a)(1) + 1 2   25a 2  – 10a + 1 = (5a – 1) 2   EXAMPLE 3 Factorize:  36m 2  – 49n 2 [ a 2  –b 2  = (a + b) (a - b) ] 36m 2  – 49n 2   = (6m) 2  – (7n) 2 36m 2  – 49n 2  = (6m + 7n) (6m – 7n)  EXAMPLE 4 Factorize: 4x 2  + 9y 2  + 25z 2  + 12xy + 30yz + 20xz [(a + b + c) 2  = a 2  + b 2  + c 2  + 2ab +2bc +2ca ]  ...

  FACTORIZATION - USING PRODUCT AND SUM EXAMPLE 1 Factorize: X 2  + 8x + 15   a= 1, b = 8, c = 15   Product = a × c                = 1 × 15                 = 15 Sum = b          = 8   To get the product 15 we can multiply 3 and 5 and to get the sum of 15 we can add 3 and 5. Therefore, the middle term 8x can be written as 3x + 5x.   X 2  + 8x + 15 =  X 2  + 3x + 5x + 15                      = (x 2  + 3x) + (5x + 15)                       = x(x + 3) + 5(x + 3)                 ...

  ALGEBRAIC IDENTITIES An identity is an equality that remains true regardless of the values chosen for its variables. Algebraic Identities     (a+b) 2  = a 2  + 2ab + b 2 (a-b) 2  = a 2  - 2ab + b 2   (a+b)(a-b) = a 2  – b 2   (x+a)(x+b) = x 2  + (a+b)x +ab 1. Prove (a+b) 2  = a 2  + 2ab + b 2   Solution:    (a+b)  2  = (a+b) (a+b)    (a+b)  2  = a (a+b) + b (a+b)    (a+b)  2  = a 2  + ab + ab + b 2          (a+b)  2  = a 2  +2ab + b 2 EXAMPLE:1  EXPAND ( 3x + 4y) 2   Solution: a =  3x and b =  4y = (3x) 2  + 2(3x)(4y) + (4y) 2 = 9x 2  + 24xy + 16y 2 2. Prove  (a-b) 2  = a 2  - 2ab + b 2 Solution: (a - b) 2   = (a-b) (a-b) (a - b) 2   = a(a-b) – b(a-b) (a - b) 2   =  a 2  – ab – ab + b 2 (a - b) 2   =  a 2  - 2ab + ...

  ALGEBRA - POLYNOMIALS TYPES OF POLYNOMIALS 1) POLYNOMIALS BASED ON TERMS MONOMIAL :  The expression which contains only one term is called as monomial.   Example: 5a, 3x BINOMIAL :  The expression which contains two terms is called as binomial. Example: 5x+3, 3a+2b TRINOMIAL :  The expression which contains only three terms is called as trinomial. Example: 4X 2 +2X+5 POLYNOMIAL :  The expression which contains two or more that many terms are called as polynomials 2) POLYNOMIALS BASED ON DEGREE CONSTANT :  A polynomial of degree zero is called as constant polynomial.  Example: 5, 7 LINEAR :  A polynomial of degree one is called as linear polynomial. Example: 4x, 7a QUADRATIC :  A polynomial of degree two is called as quadratic polynomial. Example: 4X 2 +2X+5 CUBIC :  A polynomial of degree three is called as cubic polynomial. Example: 3x 3 - 4X 2 +2X+5 ARITHMETIC OF POLYNOMIAL ADDITION OF POLYNOMIAL The addition...