The Chemistry Maths Book, Second Edition

30 Chapter 1Numbers, variables, and units Section 1.7 Find the sum and product of the pairs of complex numbers: 78.z 1 1 = 131 + ...

2 Algebraic functions 2.1 Concepts When the equation of state (1.1) of the ideal gas is written in the form it is implied that t ...

32 Chapter 2Algebraic functions EXAMPLE 2.2Replace the variable xin (2.2) by the variable a. f(a) 1 = 12 a 2 1 − 13 a 1 + 11 0 E ...

2.2 Graphical representation of functions 33 For each value of xthere exists a value of y. A table can be drawn, such as Table 2 ...

34 Chapter 2Algebraic functions A frame of referenceis defined in the plane, consisting of a fixed point called the originof the ...

2.3 Factorization and simplification of expressions 35 The inverse operation of factorization is usually called expansionor mult ...

36 Chapter 2Algebraic functions Therefore x 4 1 − 15 x 2 1 + 141 = 1 (x 2 1 − 1 1)(x 2 1 − 1 4) Both the quadratic factors have ...

2.4 Inverse functions 37 both numerator and denominator have the common factor x, and can be divided by this factor (whenx 1 ≠ 1 ...

38 Chapter 2Algebraic functions To solve for x, (i) multiply both sides of the equation by(cx 1 + 1 d): (cx 1 + 1 d)y 1 = 1 ax 1 ...

2.4 Inverse functions 39 In physical applications it is usually obvious from the context which value is to be chosen. It is also ...

40 Chapter 2Algebraic functions EXAMPLE 2.12The van der Waals equation The equation of state for a ‘slightly imperfect gas’ is ( ...

Only real coefficients are discussed here; the case of complex coefficients is shown in Section 8.4 to involve no new principles ...

42 Chapter 2Algebraic functions and (2.13) defines the constant slope. The line crosses the horizontal x-axis at one point: y 1 ...

2.5 Polynomials 43 Degree n 1 = 12 : quadratic function f(x) 1 = 1 a 0 1 + 1 a 1 x 1 + 1 a 2 x 2 (2.16) The quadratic function i ...

44 Chapter 2Algebraic functions The two roots are (2.21) and the quadratic has factorized form ax 2 1 + 1 bx 1 + 1 c 1 = 1 a(x 1 ...

2.5 Polynomials 45 EXAMPLE 2.17Case The quadratic 2 x 2 1 − 18 x 1 + 181 = 1 2(x 1 − 1 2) 2 has zero discriminant and the double ...

46 Chapter 2Algebraic functions (2.24) EXAMPLE 2.19Behaviour of a quadratic for large values of the variable For values of|x|lar ...

2.5 Polynomials 47 where vis the velocity of the body. The expression for the energy is a quadratic function of the variables va ...

48 Chapter 2Algebraic functions once. In general it has an odd number of real roots. A polynomial of even degree (n 1 = 1 2, 4, ...

2.5 Polynomials 49 Thena 1 = 11 ,b 1 = 1 − 6 ,c 1 = 110 , and the quadratic factor is x 2 1 − 16 x 1 + 1 10 with roots (see also ...