Introduction to Algebra 2 Introduction to Algebra Click on the computer image at the bottom right for a direct web link to an interesting Wikipedia Math Site. Prepared by: Richard Mitchell Humber College

CASE STUDY

2.1-Algebraic Expressions

2.1-DEFINITIONS-Pages 52 to 55 See WileyPLUS Glossary for Terms and Definitions

2.2-ADDITION AND SUBTRACTION OF ALGEBRAIC EXPRESSIONS

2.2-EXAMPLE 20-Page 56

2.2-EXAMPLE 24(c)-Page 57

2.2-EXAMPLE 25-Page 57

2.3-Integral Exponents

2.3-EXAMPLE 28(c)-Page 60

2.3-EXAMPLE 29(b) and 29(c)-Page 60

2.3-EXAMPLE 30(c) and 30(d)-Page 61

2.3-EXAMPLE 31(c) and 31(e)-Page 61

2.3-EXAMPLE 32(d) and 32(e)-Page 62

2.3-EXAMPLE 33(e)-Page 63

2.3-EXAMPLE 34(d)-Page 63

2.3-EXAMPLE extra

2.3-EXAMPLE 35(c) and 35(e)-Page 63

2.3-EXAMPLE 36(f)-Page 64

2.3-SUMMARY-Page 64 The Laws of Exponents

2.4-MULTIPLICATION OF ALGEBRAIC EXPRESSIONS

2.4-EXAMPLE 37(c)-Page 66 Multiply the following: (-p)(-q)(-r)(-s) ANS: pqrs

2.4-EXAMPLE 40-Page 67 Multiply the following: (4a3b)(-3ab2) = (4)(-3)(a3)(a1)(b1)(b2) = -12a(3+1)b(1+2) = -12a4b3 ANS: -12a4b3

2.4-EXAMPLE 41-Page 67 Multiply 5x2, -3xy, and –xy3z = (5x2)(-3xy)(-xy3z) = (5x2)(-3x1y1)(-x1y3z1) = +15x(2+1+1)y(1+3) z1 = +15x4y4z1 ANS: 15x4y4z

2.4-EXAMPLE 42-Page 67 Multiply 3x2 by 2axn = (3x2)(2axn) = (3)(2)(a)(x2)(xn) = 6ax(2+n) ANS: 6axn+2

2.4-EXAMPLE 43-Page 68 Multiply the following (without brackets): 2x(x – 3x2) = (2x)(x) + (2x)(-3x2) = (2x1)(x1) + (2x1)(-3x2) = 2x2 – 6x3 ANS: 2x2 – 6x3

2.4-EXAMPLE 44(a)-Page 68 Multiply the following (without brackets): -3x3(3x2 – 2x + 4) = (-3x3)(3x2) + (-3x3)(-2x) + (-3x3)(4) = (-3x3)(3x2) + (-3x3)(-2x1) + (-3x3)(4) = -9x5 + 6x4 – 12x3 ANS: -9x5 + 6x4 – 12x3

2.4-EXAMPLE 47(b)-Page 69 Multiply the following (without brackets): 3{2[4(w – 4) – (x + 3)] – 2} – 6x = 3{2[4(w – 4) – x – 3] – 2} – 6x = 3{2[4w – 16 – x – 3] – 2} – 6x = 3{2[4w – x – 19] – 2} – 6x = 3{8w – 2x – 38 – 2} – 6x = 3{8w – 2x – 40} – 6x = 24w – 6x – 120 – 6x = 24w – 12x – 120 ANS: 24w – 12x – 120

2.4-EXAMPLE 48-Page 69 Multiply the following (without brackets): (x + 4)(x – 3) = (x)(x) + (x)(-3) + (4)(x) + (4)(-3) = x2 – 3x + 4x – 12 = x2 + x – 12 ANS: x2 + x – 12 NB: Also known as the F.O.I.L. Rule

2.4-EXAMPLE 50(a)-Page 69 Multiply the following (without brackets): (x – 3)(x2 – 2x + 1) = (x)(x2) + (x)(-2x) + (x)(1) + (-3)(x2) + (-3)(-2x) + (-3)(1) = x3 – 2x2 + x – 3x2 + 6x - 3 = x3 – 5x2 + 7x – 3 ANS: x3 – 5x2 + 7x – 3

2.4-EXAMPLE 50(b)-Page 69 Multiply the following (without brackets): (2x + 3y – 4z)(x – 2y – 3z) = (2x)(x)+(2x)(-2y)+(2x)(-3z)+(3y)(x)+(3y)(-2y )+(3y)(-3z) +(-4z)(x)+(-4z)(-2y)+(-4z)(-3z) = 2x2 – 4xy – 6xz + 3xy – 6y2 – 9yz – 4xz + 8yz + 12z2 = 2x2 – 6y2 + 12z2 – xy – 10xz - yz ANS: 2x2 – 6y2 + 12z2 – xy – 10xz – yz

2.4-EXAMPLE 51-Page 69 ANS: 2x3 – 5x2 – x + 6 Multiply the following (without brackets): (x – 2)[(x + 1)(2x – 3)] = (x – 2)[(x)(2x) + (x)(-3) + (1)(2x) + (1)(-3)] = (x – 2)[2x2 – 3x + 2x – 3] = (x – 2)[2x2 – x – 3] = (x)(2x2) + (x)(-x) + (x)(-3) + (-2)(2x2) + (-2)(-x) + (-2)(-3) = 2x3 – x2 – 3x – 4x2 + 2x + 6 = 2x3 – 5x2 – x + 6 ANS: 2x3 – 5x2 – x + 6

2.4-EXAMPLE 52-Page 70 ANS: x3 – 3x2 + 3x – 1 Multiply the following (without brackets): (x – 1)3 = (x – 1)[(x – 1)(x – 1)] = (x – 1)[(x)(x) + (x)(-1) + (-1)(x) + (-1)(-1)] = (x – 1)[x2 – 2x + 1] = (x)(x2) + (x)(-2x) + (x)(1) + (-1)(x2) + (-1)(-2x) + (-1)(1) = x3 – 2x2 + x – x2 + 2x – 1 = x3 – 3x2 + 3x – 1 ANS: x3 – 3x2 + 3x – 1

2.4-EXAMPLES-Pages 70 to 71 Special Cases: Tricks: (x + 2)(x – 2) = x2 – 22 (x + 2)2 = (x + 2)(x + 2) = x2 + 4x + 4 (x – 2)2 = (x – 2)(x – 2) = x2 – 4x + 4 (x – 3)3 = (x – 3)(x – 3)(x – 3) = x3 – 9x2 + 27x – 27 Tricks: Perfect Square Trinomials 9x2 – 12x + 4 = (3x – 2)2

2.5-Division of Algebraic Expressions

2.5-EXAMPLE 63(a)-Page 76

2.5-EXAMPLE 64-Page 76

2.5-EXAMPLE 66-Page 76

2.5-EXAMPLE 67-Page 77

2.5-EXAMPLE 68-Page 77

2.5-EXAMPLE 69-Page 77

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