Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. Factors and Factoring Prepared by: Richard.

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Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. Factors and Factoring Prepared by: Richard Mitchell Humber College C8

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.1-Common Factors

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.1-EXAMPLE 6(a)-Page 193 Factor the following expression: 3x 3 – 2x 1 + 5x 4 = x 1 (3x 2 – 2 + 5x 3 )

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.1-EXAMPLE 6(b)-Page 193 Factor the following expression: 3x 1 y 2 – 9x 3 y 1 + 6x 2 y 2 = 3x 1 y 1 (y 1 – 3x 2 + 2x 1 y 1 )

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.1-EXAMPLE 6(c)-Page 193 Factor the following expression: 3x 3 – 6x 2 y 1 + 9x 4 y 2 = 3x 2 (x 1 – 2y 1 + 3x 2 y 2 )

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.2-Difference of Two Squares

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.2-EXAMPLE 10-Page 195 Factor the following expression: 4x 2 – 9

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.2-EXAMPLE 11(a)-Page 195 Factor the following expression: y 2 – 1

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.2-EXAMPLE 11(b)-Page 195 Factor the following expression: 9a 2 – 16b 2

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.2-EXAMPLE 11(d)-Page 195 Factor the following expression: 1 – a 2 b 2 c 2

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.2-EXAMPLE 12(b)-Page 196 Factor the following expression: x 2a – y 6b

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.2-EXAMPLE 14-Page 196 Factor the following expression: (a + b) 2 – x 2

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.2-EXAMPLE 15-Page 196 Factor the following expression: a – ab 2 =a(1 – b 2 )

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.2-EXAMPLE 16-Page 196 Factor the following expression: x 4 – y 4

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.3-Factoring Trinomials

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.3-EXAMPLE 21(a)(b)-Page 198 Factor the following expressions: x 2 + 6x + 8 x 2 – 6x + 8 = (x + 4) (x + 2) = (x – 4) (x – 2)

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.3-EXAMPLE 22-Page 199 Factor the following expression: x 2 - 2x - 8 = (x - 4) (x + 2)

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.3-EXAMPLE 23-Page 199 Factor the following expression: x 2 + 2x - 15 = (x + 5) (x - 3)

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.3-EXAMPLE 24-Page 199 Factor the following expression completely: 2x 3 + 4x 2 – 30x = 2x (x 2 + 2x - 15) = 2x (x – 3)(x + 5)

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.3-EXAMPLE 27-Page 200 Factor the following expression: x 6 – x 3 – 6 = (x 3 ) 2 – (x 3 ) – 6 = (x 3 – 3)(x 3 + 2)

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.4-Factoring by Grouping

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.4-EXAMPLE 31-Page 202 Factor ab + 4a + 3b + 12 = (ab + 4a) + (3b + 12) = a (b + 4) + 3 (b + 4) = (a + 3)●(b + 4)

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. Factor ab - 4a - 3b + 12 = (ab - 4a) - (3b - 12) = a (b - 4) - 3 (b - 4) = (a - 3)●(b - 4)

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.4-EXAMPLE 32-Page 202 (option 1) Factor x 2 – y 2 + 2x + 1 = (x 2 – y 2 ) + (2x + 1) = (x + y) (x – y) + (2x + 1)

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.4-EXAMPLE 32-Page 202 (option 2) Factor x 2 – y 2 + 2x + 1 = (x 2 + 2x) + (1 – y 2 ) = x (x + 2) + (1 + y) (1 – y)

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.4-EXAMPLE 32-Page 202 (option 3) Factor x 2 – y 2 + 2x + 1 = (x 2 + 2x + 1) - (y 2 ) = (x + 1) (x + 1) – (y 2 ) = (x + 1) 2 – (y) 2 = (x y) ●(x + 1 – y)

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.5-The General Quadratic Trinomial

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.5-EXAMPLE 33-Page 203 Factor 2x 2 + 5x + 3 by ‘Trial and Error Method’ = (2x + 1)● (x + 3) = (2x - 1)● (x - 3) = (2x + 3)● (x + 1)

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. Factor 3x 2 – 16x - 12 by ‘Trial and Error Method’ = (3x + 4)● (x - 3) = (3x - 4)● (x + 3) = (3x - 2)● (x + 6) = (3x + 2)● (x - 6)

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.5-EXAMPLE 34-Page 204 Factor 3x 2 – 16x - 12 by ‘The Grouping Method’ Multiply the Leading Coefficient by the Constant Term. This gives us -36 Find two numbers whose product equals -36 AND whose sum equals the middle coefficient -16. These numbers are 2 and -18 Rewrite the trinomial, splitting the middle term according to the selected factors (i.e. -16x = 2x – 18x) then group the first two terms together and the last two terms together (see 8.4-Grouping Method). Remove Common Factors from each grouping (see 8.4). Remove the Common Factor/Brackets from the entire expression.

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. Factor 3x 2 – 16x - 12 by ‘The Grouping Method’ Multiply First and Last Terms (3)●(-12) = -36 Focus on the sum of the middle number -16 Try (-1)●(+36)= -36 but (-1) + (+36) ≠ -16 Try (+1)●(-36)= -36 but (+1) + (-36) ≠ -16 Try (-2)●(+18)= -36 but (-2) + (+18) ≠ -16 Try (+2)●(-18)= -36 AND (+2) + (-18) = -16 Rewrite the Trinomial 3x x + 2x – 12 Group the First Two and Last Two terms together. (3x 2 – 18x) + (2x - 12) Use ‘The Grouping Method’ to find Common Factors. 3x(x – 6) + 2(x – 6) = (3x + 2) (x – 6)

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.6-The Perfect Square Trinomial

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.6-EXAMPLE 36-Page 206 Square the binomial (2x + 3) = (2x + 3) 2 = (2x + 3)(2x + 3) = 4x 2 + 6x + 6x + 9 = 4x x + 9Rule: Step 1: Square each of the first and last terms. Step 2: Multiply the first and last terms. Double your answer.

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.7-Sum or Difference of Two Cubes

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. 8.7-EXAMPLE 39-Page 208 Factor x = x = (x + 3)(x 2 – 3x + 9) SUBSTITUTE: a=x and b=3 into Eqn.42

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons Canada, Ltd. Copyright Copyright © 2008 John Wiley & Sons Canada, Ltd. All rights reserved. Reproduction or translation of this work beyond that permitted by Access Copyright (The Canadian Copyright Licensing Agency) is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons Canada, Ltd. The purchaser may make back-up copies for his or her own use only and not for distribution or resale. The author and the publisher assume no responsibility for errors, omissions, or damages caused by the use of these programs or from the use of the information contained herein.