Algebra - CHAPTER 1 Review of Algebra Much of the material...

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Unformatted text preview: CHAPTER 1 Review of Algebra Much of the material in this chapter is revision from GCSE maths (al- though some of the exercises are harder). Some of it – particularly the work on logarithms – may be new if you have not done A-level maths. If you have done A-level, and are confident, you can skip most of the ex- ercises and just do the worksheet, using the chapter for reference where necessary. — ./ — 1. Algebraic Expressions 1.1. Evaluating Algebraic Expressions Examples 1.1 : (i) A firm that manufactures widgets has m machines and employs n workers. The number of widgets it produces each day is given by the expression m 2 ( n- 3). How many widgets does it produce when m = 5 and n = 6? Number of widgets = 5 2 × (6- 3) = 25 × 3 = 75 (ii) In another firm, the cost of producing x widgets is given by 3 x 2 + 5 x + 4. What is the cost of producing (a) 10 widgets (b) 1 widget? When x = 10, cost = (3 × 10 2 ) + (5 × 10) + 4 = 300 + 50 + 4 = 354 When x = 1, cost = 3 × 1 2 + 5 × 1 + 4 = 3 + 5 + 4 = 12 It might be clearer to use brackets here, but they are not essential: the rule is that × and ÷ are evaluated before + and- . (iii) Evaluate the expression 8 y 4- 12 6- y when y =- 2. (Remember that y 4 means y × y × y × y .) 8 y 4- 12 6- y = 8 × (- 2) 4- 12 6- (- 2) = 8 × 16- 12 8 = 128- 1 . 5 = 126 . 5 (If you are uncertain about using negative numbers, work through Jacques pp.7–9.) Exercises 1.1 : Evaluate the following expressions when x = 1, y = 3, z =- 2 and t = 0: (a) 3 y 2- z (b) xt + z 3 (c) ( x + 3 z ) y (d) y z + 2 x (e) ( x + y ) 3 (f) 5- x +3 2 t- z 1 2 1. REVIEW OF ALGEBRA 1.2. Manipulating and Simplifying Algebraic Expressions Examples 1.2 : (i) Simplify 1 + 3 x- 4 y + 3 xy + 5 y 2 + y- y 2 + 4 xy- 8. This is done by collecting like terms , and adding them together: 1 + 3 x- 4 y + 3 xy + 5 y 2 + y- y 2 + 4 xy- 8 = 5 y 2- y 2 + 3 xy + 4 xy + 3 x- 4 y + y + 1- 8 = 4 y 2 + 7 xy + 3 x- 3 y- 7 The order of the terms in the answer doesn’t matter, but we often put a positive term first, and/or write “higher-order” terms such as y 2 before “lower-order” ones such as y or a number. (ii) Simplify 5( x- 3)- 2 x ( x + y- 1). Here we need to multiply out the brackets first, and then collect terms: 5( x- 3)- 2 x ( x + y- 1 = 5 x- 15- 2 x 2- 2 xy + 2 x = 7 x- 2 x 2- 2 xy + 5 (iii) Multiply x 3 by x 2 . x 3 × x 2 = x × x × x × x × x = x 5 (iv) Divide x 3 by x 2 . We can write this as a fraction, and cancel: x 3 ÷ x 2 = x × x × x x × x = x 1 = x (v) Multiply 5 x 2 y 4 by 4 yx 6 . 5 x 2 y 4 × 4 yx 6 = 5 × x 2 × y 4 × 4 × y × x 6 = 20 × x 8 × y 5 = 20 x 8 y 5 Note that you can always change the order of multiplication. (vi) Divide 6 x 2 y 3 by 2 yx 5 ....
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This note was uploaded on 04/12/2010 for the course ECON DEAM taught by Professor Vines during the Spring '10 term at Oxford University.

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Algebra - CHAPTER 1 Review of Algebra Much of the material...

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