JMATHSLEARNING

Formulae for binomial     1.        (a + b) 2 = a 2  + 2ab + b 2 Proof :    Taking L.H.S              (a + b) 2 = (a + b) . (a + b)                           = a (a + b ) + b (a + b)                           = a 2 + ab + ab + b 2            (a + b) 2   = a 2  + 2ab + b 2       …….. (i)     2.        (a – b) 2 = a 2 – 2ab + b 2 Proof :  Putting b = – b in equ n  (i)         (a – b) 2   = a 2  + 2a(–b) + (– b) 2         (a – b) 2 = a 2   –  2ab + b 2     3.        a 2 – b 2 = (a + b) . (a – b) Proof :   Taking R.H.S        (a + b) . (a – b) = a (a – b) + b (a – b)                                  = a 2 –  ab + ab – b 2                                  = a 2 – b 2    4.        (a + b) 3 = a 3 + b 3 + 3ab (a + b) = a 3 + b 3 + 3a 2 b + 3ab 2 Proof : Taking L.H.S    (a + b) 3 = (a + b) . (a + b) 2                 = (a + b) . (a 2  + 2ab + b 2 )                 = a (a 2  + 2ab + b 2 ) + b (a 2  + 2ab + b 2 )                 = a

Remainder Theorem Let p(x) be a polynomial of degree   ≥  1  and let a be any real number. When  p(x) is divided by ( x – a), then the remainder is p(a). Proof : Suppose that when p(x) is divided by (x – a), the quotient is q(x) and the remainder is r.           ∴          Dividend = Divisor x Quotient + Remainder                                 p(x) = (x – a) . q(x) + r        ……….. (i)                                 Putting x = a in (i),                                  We get r = p(a) Thus, when p(x) is divided by (x – a), then the remainder is p(a). Example: Find the remainder when the polynomial                  p(x) = x 3 + 3x 2 – 2x +1 is divided by (x – 4). Solution:        p(x) = x 3 + 3x 2 – 2x +1                            x – 4 = 0  ⇒ x = 4               By the remainder theorem, we that when p(x) is divided by (x – 4), the remainder is p(4). Now, remainder = p(4)                               = (4 3 + 3 x 4 2 – 2 x 4 + 1)       

Constant A symbol having a fixed numerical value is called a constant Examples:  3, -6, 5/6, 0.23, π, etc. are all constants. Variable A symbol which may be assigned different numerical values us known as a variable . Example: We know that the area of a circle is given by the formula A = πr 2 , where r is the radius of the circle. Here, π is constant, while A and r are variables. Algebraic Expression A combination of constants and variables, connected by some or all of the operations +, − , × , and ÷ is known an algebraic expression . Examples: x +2, x 2 +y 2 , x – 1/x, etc. Terms of an Algebraic Expression The several parts of an algebraic expression separated by + or – operations are called the terms of the expression . Examples:   (i). 3 + 4x – 7xy 2 + 9xy is an algebraic expression containing four terms , namely, 3, 4x, 7xy 2 and 9xy     (ii). X 3 + 3x 2 y – 5xy 2 + y 3 – 6 is an algebraic expression containing five terms , namely, X 3 , 3x