64 x x M1 2 M1 for identifying the two terms M Where two terms are and + 1 the expression required is: A1 A1 for correct expression as asked x (x + + 1)2 (x2 + (x + 1)2) – x2 = (2x + 1)2 (x2 + + 2x + 1) – = 4x2 + 4x + 1 2x2 2x 1 – – – A1 A1 for correct expansion of all brackets A1 A1 for correct simplification = 2x2 + 2x A1 A1 for correct factorisation = 2x(x + 1) …..(1) B1 B1 for clear complete proof with correct mathematical x(x Where Tx = + 1) notation 4Tx = 2x(x + 1), which is same as the 6 result in equation (1). 65 p x 2 M Let = B1 q B1 for expressed algebraically q x 1 Then = + B1 r r x B1 for expressed algebraically and = + 2 B1 pr pr x(x B1 for product expressed algebraically so = + 2) B1 q2 B1 for 1 expressed algebraically x2 B1 – = + 2x B1 for simplification B1 q2 1 = (x + 1)2 1 – – B1 for complete clear proof x2 6 = + 2x + 1 1 – x2 pr = + 2x = 66 a There are many equivalent B1 2 B1 for a correct example H expressions. 3 b i For example, expand the bracketed q2 q B1 B1 for a correct example term: 4 – – – For example: For example: 4x2 + 10x ii B1 B1 for a correct example 3 67 a To make it a product of two linear B1 2 B1 for clear explanation H expressions. 3 b The quadratic expression. B1 B1 for clear explanation c That the signs of the numbers in the B1 B1 for clear explanation brackets are different. d One factor of the constant term is zero. B1 B1 for clear explanation There is only one set of brackets. 4
64 Where two terms are x and x + 1 the expression required is: (x + x + 1)2 – (x2 + (x + 1)2) = (2x + 1)2 – (x2 + x2 + 2x + 1) = 4x2 + 4x + 1 – 2x2 – 2x – 1 = 2x2 + 2x = 2x(x + 1) …..(1) Where Tx = x(x + 1) 4Tx = 2x(x + 1), which is same as the result in equation (1). M1 A1 A1 A1 A1 B1 2 M1 for identifying the two terms A1 for correct expression as asked A1 for correct expansion of all brackets A1 for correct simplification A1 for correct factorisation B1 for clear complete proof with correct mathematical notation M 6 65 Let p = x Then q = x + 1 and r = x + 2 so pr = x(x + 2) = x2 + 2x q2 – 1 = (x + 1)2 – 1 = x2 + 2x + 1 – 1 = x2 + 2x = pr B1 B1 B1 B1 B1 B1 2 B1 for q expressed algebraically B1 for r expressed algebraically B1 for product pr expressed algebraically B1 for q2 – 1 expressed algebraically B1 for simplification B1 for complete clear proof M 6 66 a b i ii There are many equivalent expressions. For example, expand the bracketed term: – q2 – q – 4 For example: For example: 4x2 + 10x B1 B1 B1 2 3 B1 for a correct example B1 for a correct example B1 for a correct example H 3 67 a b c d To make it a product of two linear expressions. The quadratic expression. That the signs of the numbers in the brackets are different. One factor of the constant term is zero. There is only one set of brackets. B1 B1 B1 B1 2 3 B1 for clear explanation B1 for clear explanation B1 for clear explanation B1 for clear explanation H 4
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