Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons...

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

Factors and Factoring

Prepared by: Richard Mitchell Humber College

C8


8.1-Common Factors8.1-Common Factors


8.1-EXAMPLE 6(a)-Page 1938.1-EXAMPLE 6(a)-Page 193Factor the following expression:

3x3 – 2x1 + 5x4

= x1 (3x2 – 2 + 5x3)

2 3ANS: (3 2 5 )x x x


8.1-EXAMPLE 6(b)-Page 1938.1-EXAMPLE 6(b)-Page 193Factor the following expression:

3x1y2 – 9x3y1 + 6x2y2

= 3x1y1 (y1 – 3x2 + 2x1y1)

2ANS: 3 ( - 3 2 )xy y x xy


8.1-EXAMPLE 6(c)-Page 1938.1-EXAMPLE 6(c)-Page 193Factor the following expression:

3x3 – 6x2y1 + 9x4y2

= 3x2 (x1 – 2y1 + 3x2y2)

2 2 2ANS: 3 ( 2 3 ) x x y x y


8.2-Difference of Two 8.2-Difference of Two SquaresSquares


8.2-EXAMPLE 10-Page 1958.2-EXAMPLE 10-Page 195Factor the following expression:

4x2 – 9

ANS: (2 3) (2 3)x x -

2 22 3x 2 232x

32 32x x 32 32x x 32 2 3x x


8.2-EXAMPLE 11(a)-Page 1958.2-EXAMPLE 11(a)-Page 195Factor the following expression:

y2 – 1

ANS: ( 1) ( 1)y y -

2 21y 2 21y

1 1y y 1 1y y 1 1y y



9a2 – 16b2

ANS: (3 4 ) (3 4 )a b a - b

2 23 4a b 2 23 4ba

43 43b ba a 43 43b ba a 43 43b ba a


8.2-EXAMPLE 11(d)-Page 1958.2-EXAMPLE 11(d)-Page 195Factor the following expression:

1 – a2b2c2

ANS: (1 ) (1 )abc - abc

2 21 abc 2 21 abc

1 1abc abc 1 1abc abc 1 1abc abc



x2a – y6b

3 3ANS: ( ) ( )a b a bx y x - y

232a bx y 232a bx y

3 3ba a by yx x 3 3ba a by yx x 3 3ba a by yx x



(a + b)2 – x2

ANS: ( ) ( )a b x a b x

2 2a b x 2 2a xb

xa b a b x xa b a b x xa b a xb



a – ab2

=a(1 – b2)

ANS: (1 ) (1 )a b - b

2 21 ba 2 21 ba

1 1a b b 1 1a b b 1 1a b b



x4 – y4

2 2ANS: ( )( )( )x y x y x y

2 22 2x y 2 22 2x y

22 2 2x xy y 22 2 2x xy y 22 2 2x xy y

2 2 x yx y x y


8.3-Factoring Trinomials8.3-Factoring Trinomials


8.3-EXAMPLE 21(a)(b)-Page 8.3-EXAMPLE 21(a)(b)-Page 198198

Factor the following expressions:

x2 + 6x + 8 x2 – 6x + 8

= (x + 4) (x + 2) = (x – 4) (x – 2)

ANS: ( 4)( 2)x x ANS: ( 4)( 2)x x



x2 - 2x - 8

= (x - 4) (x + 2)

ANS: ( 4)( 2)x x



x2 + 2x - 15

= (x + 5) (x - 3)

ANS: ( 5)( 3)x x


8.3-EXAMPLE 24-98.3-EXAMPLE 24-Page 199Factor the following expression completely:

2x3 + 4x2 – 30x

= 2x (x2 + 2x - 15)

= 2x (x – 3)(x + 5)

ANS: 2 ( 3)( 5)x x x


8.3-EXAMPLE 27-08.3-EXAMPLE 27-Page 200Factor the following expression:

x6 – x3 – 6

= (x3)2 – (x3) – 6

= (x3 – 3)(x3 + 2)

3 3ANS: ( 3)( 2)x x


8.4-Factoring by Grouping8.4-Factoring by Grouping


8.4-EXAMPLE 31-Page 2028.4-EXAMPLE 31-Page 202Factor ab + 4a + 3b + 12

= (ab + 4a) + (3b + 12)

= (ab + 4a) + (3b + 12)

= a (b + 4) + 3 (b + 4)

= (a + 3)●(b + 4)

ANS: ( 3)( 4)a b


8.4-EXAMPLE 8.4-EXAMPLE extraextra

Factor ab - 4a - 3b + 12

= (ab - 4a) - (3b - 12)

= (ab - 4a) - (3b - 12)

= a (b - 4) - 3 (b - 4)

= (a - 3)●(b - 4)

ANS: ( 3)( 4)a b


8.4-EXAMPLE 32-Page 2028.4-EXAMPLE 32-Page 202 (option (option 1)1)

Factor x2 – y2 + 2x + 1

= (x2 – y2 ) + (2x + 1)

= (x + y) (x – y) + (2x + 1)

ooooooop's


8.4-EXAMPLE 32-Page 202 8.4-EXAMPLE 32-Page 202 (option (option 2)2)

Factor x2 – y2 + 2x + 1

= (x2 + 2x) + (1 – y2)

= x (x + 2) + (1 + y) (1 – y)

ooooooop's


8.4-EXAMPLE 32-Page 202 8.4-EXAMPLE 32-Page 202 (option (option 3)3)

Factor x2 – y2 + 2x + 1

= (x2 + 2x + 1) - (y2)

= (x + 1) (x + 1) – (y2)

= (x + 1)2 – (y)2

= (x + 1 + y) ●(x + 1 – y)

ANS: ( 1 )( 1 )x y x y


8.5-The General Quadratic 8.5-The General Quadratic TrinomialTrinomial


8.5-EXAMPLE 33-Page 2038.5-EXAMPLE 33-Page 203Factor 2x2 + 5x + 3 by ‘Trial and Error Method’

= (2x + 1)● (x + 3)

= (2x - 1)● (x - 3)

= (2x + 3)● (x + 1)

ANS: (2 3)( 1)x x

ooooooo'ps

OK

ooooooo'ps


8.5-EXAMPLE 34-Page 2048.5-EXAMPLE 34-Page 204extraextra

Factor 3x2 – 16x - 12 by ‘Trial and Error Method’

= (3x + 4)● (x - 3)

= (3x - 4)● (x + 3)

= (3x - 2)● (x + 6)

= (3x + 2)● (x - 6)

ANS: (3 2)( 6)x x

ooooooo'ps

OK

ooooooo'ps

ooooooo'ps


8.5-EXAMPLE 34-Page 2048.5-EXAMPLE 34-Page 204 Factor 3x2 – 16x - 12 by ‘The Grouping Method’

Multiply the Leading Coefficient by the Constant Term. Multiply the Leading Coefficient by the Constant Term. This gives us -36This gives us -36

Find two numbers whose product equals -36 Find two numbers whose product equals -36 ANDAND whose sum whose sum equals the middle coefficient equals the middle coefficient -16-16. These numbers are . These numbers are 22 and and -18-18

Rewrite the trinomial, splitting the middle term according to the Rewrite the trinomial, splitting the middle term according to the selected factors (i.e. selected factors (i.e. -16-16x = x = 22x x –– 1818x) then group the first two termsx) then group the first two termstogether and the last two terms together (see 8.4-Grouping Method).together and the last two terms together (see 8.4-Grouping Method).

Remove Common Factors from each grouping (see 8.4).Remove Common Factors from each grouping (see 8.4).

Remove the Common Factor/Brackets from the entire expression.Remove the Common Factor/Brackets from the entire expression.

ANS: (3 2)( 6)x x


8.5-EXAMPLE 34-Page 2048.5-EXAMPLE 34-Page 204extraextra

Factor 3x2 – 16x - 12 by ‘The Grouping Method’

Multiply First and Last TermsMultiply First and Last Terms(3)(3)●(-12) = -36●(-12) = -36

Focus on the sum of the middle number Focus on the sum of the middle number -16-16Try (-1)●(+36)= -36 but (-1) + (+36) ≠ -16 Try (-1)●(+36)= -36 but (-1) + (+36) ≠ -16 Try (+1)●(-36)= -36 but (+1) + (-36) ≠ -16Try (+1)●(-36)= -36 but (+1) + (-36) ≠ -16Try (-2)●(+18)= -36 but (-2) + (+18) ≠ -16 Try (-2)●(+18)= -36 but (-2) + (+18) ≠ -16 Try (Try (+2+2)●()●(-18-18)= -36 AND ()= -36 AND (+2+2) + () + (-18-18) = ) = -16-16

Rewrite the TrinomialRewrite the Trinomial3x3x22 - 18- 18x x + 2+ 2x – 12x – 12

Group the First Two and Last Two terms together.Group the First Two and Last Two terms together.(3x(3x22 – 18– 18x) x) ++ ( (22x - 12)x - 12)

Use ‘The Grouping Method’ to find Common Factors.Use ‘The Grouping Method’ to find Common Factors.3x(x – 6) + 2(x – 6) = (3x + 2) (x – 6)3x(x – 6) + 2(x – 6) = (3x + 2) (x – 6) ANS: (3 2)( 6)x x


8.6-The Perfect Square 8.6-The Perfect Square TrinomialTrinomial


8.6-EXAMPLE 36-Page 2068.6-EXAMPLE 36-Page 206Square the binomial (2x + 3)

= (2x + 3)2

= (2x + 3)(2x + 3)

= 4x2 + 6x + 6x + 9

= 4x2 + 12x + 9

2ANS: 4 12 9x x

2(2 )3x

Rule:Rule:

2 124 9x x

Step 1: Square each of the first and last terms.

2 124 9x x

Step 2: Multiply the first and last terms. Double your answer.

2(2 )3x


8.7-Sum or Difference of Two 8.7-Sum or Difference of Two CubesCubes


8.7-EXAMPLE 39-Page 2088.7-EXAMPLE 39-Page 208Factor x3 + 27

= x3 + 33

= (x + 3)(x2 – 3x + 9)

2ANS: ( 3)( 3 9)x x x

SUBSTITUTE:SUBSTITUTE: a=x and b=3 into Eqn.42


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.

Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons...

Documents

Transcript of Calter & Calter, Technical Mathematics with Calculus, Canadian Edition ©2008 John Wiley & Sons...