Core Mathematics 3 Algebra

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1 Core Mathematics 3 Algebra Edited by K V Kumaran Core Maths 3 Algebra Page

2 Algebra fractions C3 The specifications suggest that you should be able to do the following: Simplify rational expressions including factorising, and cancelling, and algebraic division Algebraic fractions are covered at GCSE and this should only need refreshing Additions and subtraction in fractional expressions with different denominators Step : Factorise the denominators Step : Find the LCM of the denominators Step 3: Write all the fractions with the common denominators Step 4: Simplify the numerator Step 5: Factorise the numerator (If possible) Step 6: Simplify cancelling the common factors in both the denominator and numerator Core Maths 3 Algebra Page

3 Example Express y 8 y 3 (y 3)(y ) (y )(y 8) As a single fraction The factor (y + ) is common to both fractions, therefore the common denominator must be (y + ) (y + 3) (y + 8) Therefore the expression becomes: y 8 y 3 (y 3)(y ) (y )(y 8) (y 8)(y 8) (y 3)(y 3) (y 3)(y )(y 8) y 6y 64 y 6y 9 y 6y 64 y 6y 9 (y 3)(y )(y 8) 0y 55 (y 3)(y )(y 8) ) Core Maths 3 Algebra Page 3

4 Example Express 4 5 (x ) x 8x 9 As a single fraction Hence solve 4 5 (x ) x 8x 9 There is a hint in the question The common denominator must be (x - ) is one of its factors The other factor is (x + 9) Therefore the algebraic fraction becomes: x 8x 9 as 4 5 (x ) x 8x 9 4(x 9) 5 x 8x 9 4x 5 x 8x 9 Core Maths 3 Algebra Page 4

5 Hence solve 4 5 (x ) x 8x 9 4x 5 x 8x 9 Cross multiplying gives: 4x 5 x 8x 9 x 4x 60 0 (x 0)(x 6) 0 x 6 or -0 Edexcel-C3 Rational expressions past paper questions Core Maths 3 Algebra Page 5

6 Express x 3x (x 3)( x ) x 6 x (7) (Q Jan 006) 3x x (a) Simplify x (b) Hence, or otherwise, express form 3x x x x( x ) (3) as a single fraction in its simplest (3) (Q June 006) 3 Given that 4 x 3x x (ax + bx + c) + ( x ) find the values of the constants a, b, c, d and e dx e, ( x ) (Q Jan 008) 4 The function f is defined by f: x ( x ) x x 3, x > 3 x 3 (a) Show that f(x) =, x > 3 x (Q4 June 008) Core Maths 3 Algebra Page 6

7 5 f(x) = x x x 3 x x 3 (a) Express f(x) (Q Jan 009) 6 The function f is defined by f(x) = ( x 4) + x 8, x R, x 4, x ( x )( x 4) (a) Show that f (x) = x 3 x (5) (Q7 June 009) 7 Express x 3x 3 3x (Q Jan 00) 8 (a) Simplify fully x 9x 5 x x 5 (3) (Q8 June 00) 9 (a) Express Core Maths 3 Algebra Page 7

8 4x ( x ) 3 ( x )(x ) Given that f(x) = 4x ( x ) 3 ( x )(x ), x >, (b) show that f(x) = 3 x () (Q Jan 0) 0 f(x) = 4x 5 (x )( x 3) x, x 3, x x 9 (a) Show that f(x) = 5 (x )( x 3) (5) (Q7 June 0) The function f is defined by f : x 3( x ) x 7x 4, x R, x > x 4 (a) Show that f(x) = x (Q7 Jan 0) Express (3x ) 9x 4 3x (Q June 0) Core Maths 3 Algebra Page 8

9 3 h(x) = x + 4 x 5 ( x 8, x 0 5)( x ) (a) Show that h(x) = x x 5 (Q7 Jan 03) 4 Given that 4 3 ax bx c 3x x 5x 4 dx e x 4 x 4, x find the values of the constants a, b, c, d and e (Q June 03) 5 Express 3x 5 x x x 3 (Q June 03_R) 6 g( x) x 3 x x 3 x x 6, x > 3 (a) Show that x g( x), x > 3 x (Q5 June 04) 7 Express 3 6 x3 x3 4x 9 (Q June 04_R) Core Maths 3 Algebra Page 9

10 8 5 x 4 f ( x) 3x 4 x 3x 4 x, x > (a) Express f(x) (Q Jan 04_R*) 9 Given that 4x 3 + x + 7x + 8 (Ax + B)(x + 4) + Cx + D (a) find the values of the constants A, B, C and D (Q3 Jan 04_IAL) 0 Given that k is a negative constant and that the function f(x) is defined by ( x 5k)( x k) f (x) =, x 0, x 3kx k (a) show that f (x) = x k x k (3) (Q9 June 05) f(x) = (a) Given that x 4 3 x 3x 7x 6, x >, x R x x 6 x 4 x 3 3 x 7 x 6 B x A x x 6 x, find the values of the constants A and B (Q6 June 06*) Express 4x x x + 3 (Q June 07) Core Maths 3 Algebra Page 0

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