Chapter 1 More about Factorization of Polynomials 1A p.2 1B p.9 1C p.17 1D p.25 1E p.32 Chapter 2 Laws of Indices 2A p.39 2B p.49 2C p.57 2D p.68 Chapter 3 Percentages (II) 3A p.74 3B p.83 3C p.92 3D p.99 3E p.107 3F p.119 For any updates of this book, please refer to the subject homepage: http://teacher.lkl.edu.hk/subject%20homepage/MAT/index.html For mathematics problems consultation, please email to the following address: [email protected] For Maths Corner Exercise, please obtain from the cabinet outside Room 309 1 F3A: Chapter 1A Date Task Progress ○ Complete and Checked Lesson Worksheet ○ Problems encountered ○ Skipped (Full Solution) ○ Complete Book Example 1 ○ Problems encountered ○ Skipped (Video Teaching) ○ Complete Book Example 2 ○ Problems encountered ○ Skipped (Video Teaching) ○ Complete Book Example 3 ○ Problems encountered ○ Skipped (Video Teaching) ○ Complete Book Example 4 ○ Problems encountered ○ Skipped (Video Teaching) ○ Complete Book Example 5 ○ Problems encountered ○ Skipped (Video Teaching) ○ Complete Book Example 6 ○ Problems encountered ○ Skipped (Video Teaching) ○ Complete and Checked Consolidation Exercise ○ Problems encountered ○ Skipped (Full Solution) ○ Complete and Checked Maths Corner Exercise Teacher’s ○ Problems encountered ___________ 1A Level 1 Signature ○ Skipped ( ) ○ Complete and Checked Maths Corner Exercise Teacher’s ○ Problems encountered ___________ 1A Level 2 Signature ○ Skipped ( ) 2 ○ Complete and Checked Maths Corner Exercise Teacher’s ○ Problems encountered ___________ 1A Level 3 Signature ○ Skipped ( ) ○ Complete and Checked Maths Corner Exercise Teacher’s ○ Problems encountered ___________ 1A Multiple Choice Signature ○ Skipped ( ) ○ Complete and Checked Mark: E-Class Multiple Choice ○ Problems encountered Self-Test ○ Skipped _________ 3 Book 3A Lesson Worksheet 1A (Refer to §1.1A) 1.1 Factorization Using Identities 1.1A Using the Difference of Two Squares Identity a2 – b2 ≡ (a + b)( a – b) Example 1 Instant Drill 1 Factorize Factorize 2 2 Factorize: convert the 2 2 (a) x – 3 , polynomial into (a) p – 4 , 2 2 2 2 (b) 5 – y . the product of its (b) 8 – q . factors. a = ___, b = ___ Sol (a) x2 – 32 a = x, b = 3 Sol (a) p2 – 42 = (x + 3)( x – 3) = ( )( ) a = ___, b = ___ (b) 52 – y2 a = 5, b = y (b) 82 – q2 = (5 + y)(5 – y) = ( )( ) ○○○→→→ Ex 1A 1, 2 Example 2 Instant Drill 2 Factorize Factorize 2 Do you remember 2 (a) x – 4, the square (a) h – 9, (b) y2 – 36. numbers 1, 4, 9, (b) k2 – 49. 16, 25, ? 2 4 = 2 2 9 = ( ) Sol (a) x2 – 4 Sol (a) h2 – 9 = x2 – 22 = ( ) 2 – ( ) 2 = (x + 2)( x – 2) = ( )( ) 36 = 62 (b) y2 49 = ( ) 2 – 36 (b) k2 – 49 y2 2 = – 6 = ( ) 2 ( ) 2 y y = ( + 6)( – 6) = ( )( ) 1. Factorize (a) x2 – 64, (b) u2 – 100, (c) w2 – 121. 2 10 = ___ 2 11 = ___ 4 2. Factorize (a) 25 – y2, (b) 36 – p2, (c) 81 – n2. ○○○→→→ Ex 1A 3–5 Example 3 Instant Drill 3 Factorize Factorize (a) 4x2 – 1, (a) 9x2 – 16, (b) 4x2 – y2. (b) 9x2 – 16 y2. a = ( ) Sol (a) 4x2 – 1 Sol (a) 9x2 – 16 a = 2 x b = ( ) = (2 x)2 – 12 b = 1 = ( ) 2 – ( ) 2 = (2 x + 1)(2 x – 1) = a = ( ) (b) 4x2 – y2 (b) 9x2 – 16 y2 2 2 a = 2 x 2 2b = ( ) = (2 x) – y b = y = ( ) – ( ) = (2 x + y)(2 x – y) = 3. Factorize 4. Factorize (a) 1 – 36 x2, (a) 4x2 – 25 y2, 4x2 = ( ) 2 (b) 25 p2 – 49, (b) 9h2 – 64 k2, 25 y2 = ( ) 2 (c) 64 – 81 s2. (c) 49 m2 – 100 n2. ○○○→→→ Ex 1A 8–14 5. Factorize 2 2 2 6. Factorize x y = ( xy) 2 2 (a) x2y2 (a) x2 y2z2 4x = ( ) – 16, 4 – , 2 2 2 (b) p2q2 – 81. (b) s2 – 25 p2q2. y z = ( ) 5 ○○○→→→ Ex 1A 15–18 Level Up Questions Factorize the following polynomials. [Nos. 7 −−−8] 7. (a) 49 b2 – 36 a2c2 (b) –9x2y2 + 16 w2 = = 16 w2 – ( ) = (c) 121 h2 – 144 m2n2 = 8. (a) 8x2 – 8 Take out the (b) 6x2 – 6y2 common The common factor is = 8( ) factor 8 = ( )( _________. ) = first. = B 6 New Century Mathematics (2nd Edition) 3A 1 More about Factorization of Polynomials Consolidation Exercise 1A Level 1 Factorize the following polynomials. [Nos. 1–18] 1. x2 − 22 2. 42 − y2 3. z2 − 64 4. c2 − 100 5. 36 − x2 6. −y2 + 1 7. 1 − 16 u2 8. 25 m2 − 49 9. 16 a2 − 9 10. −81 x2 + 25 11. 36 a2 − b2 12. 4p2 − 25 q2 13. c2d2 − 9 14. −49 + h2k2 15. 16 p2 − q2r2 16. x2 − 100 y2z2 17. 11 x2 − 11 18. 3m2 − 3n2 Level 2 Factorize the following polynomials. [Nos. 19–33] 19. (3 + x)2 − 1 20. (y − 3) 2 − 25 21. 121 − (m − 2n)2 22. (3 + x)2 − (1 + 2 x)2 23. (2 x + 3 y)2 − (x − 2y)2 24. (a − 2b)2 − (2 a + b)2 25. 3 − 75 x2 26. 18 c2 − 72 d2 27. 6ab 2 − 24 ac 2 28. 5( p + q)2 − 45 29. 32 h2 − 2( k − 3) 2 30. 18( x + 2 y)2 − 2( x − y)2 31. 16 − a2 + 4 b − ab 32. 2x + 7 y + 4 x2 − 49 y2 33. p2 + 3 q − 3p − q2 7 Answer Consolidation Exercise 1A 1. (x + 2)( x − 2) 2. (4 + y)(4 − y) 3. (z + 8)( z − 8) 4. (c + 10)( c − 10) 5. (6 + x)(6 − x) 6. (1 + y)(1 − y) 7. (1 + 4 u)(1 − 4u) 8. (5 m + 7)(5 m − 7) 9. (4 a + 3)(4 a − 3) 10. (5 + 9 x)(5 − 9x) 11. (6 a + b)(6 a − b) 12. (2 p + 5 q)(2 p − 5q) 13. (cd + 3)( cd − 3) 14. (hk + 7)( hk − 7) 15. (4 p + qr)(4 p − qr ) 16. (x + 10 yz )( x − 10 yz ) 17. 11( x + 1)( x − 1) 18. 3( m + n)( m − n) 19. (4 + x)(2 + x) 20. (y + 2)( y − 8) 21. (11 + m − 2n)(11 − m + 2 n) 22. (4 + 3 x)(2 − x) 23. (3x + y)( x + 5y) 24. −(3 a − b)(a + 3b) 25. 3(1 + 5 x)(1 − 5x) 26. 18( c + 2 d)( c − 2d) 27. 6a(b + 2 c)( b − 2c) 28. 5( p + q + 3)( p + q − 3) 29. 2(4 h + k − 3)(4 h − k + 3) 30. 2(4 x + 5 y)(2 x + 7y) 31. (4 − a)(4 + a + b) 32. (2 x + 7 y)(1 + 2x − 7y) 33. (p − q)( p + q − 3) 8 F3A: Chapter 1B Date Task Progress ○ Complete and Checked Lesson Worksheet ○ Problems encountered ○ Skipped (Full Solution) ○ Complete Book Example 7 ○ Problems encountered ○ Skipped (Video Teaching) ○ Complete Book Example 8 ○ Problems encountered ○ Skipped (Video Teaching) ○ Complete Book Example 9 ○ Problems encountered ○ Skipped (Video Teaching) ○ Complete Book Example 10 ○ Problems encountered ○ Skipped (Video Teaching) ○ Complete Book Example 11 ○ Problems encountered ○ Skipped (Video Teaching) ○ Complete and Checked Consolidation Exercise ○ Problems encountered ○ Skipped (Full Solution) ○ Complete and Checked Maths Corner Exercise Teacher’s ○ Problems encountered ___________ 1B Level 1 Signature ○ Skipped ( ) ○ Complete and Checked Maths Corner Exercise Teacher’s ○ Problems encountered ___________ 1B Level 2 Signature ○ Skipped ( ) ○ Complete and Checked Maths Corner Exercise Teacher’s ○ Problems encountered ___________ 1B Level 3 Signature ○ Skipped ( ) ○ Complete and Checked Maths Corner Exercise Teacher’s ○ Problems encountered ___________ 1B Multiple Choice Signature ○ Skipped ( ) 9 ○ Complete and Checked Mark: E-Class Multiple Choice ○ Problems encountered Self-Test ○ Skipped _________ 10 Book 3A Lesson Worksheet 1B (Refer to §1.1B) 1.1B Using the Perfect Square Identities (I) Square of the Sum of Two Numbers a2 + 2 ab + b2 ≡ (a + b)2 Example 1 Instant Drill 1 Factorize x2 + 2 x + 1. Factorize x2 + 16 x + 64. a = ___, b = ___ Sol x2 + 2 x + 1 a = x, b = Sol x2 + 16 x + 64 = x2 + 2( x)(1) + 1 2 1 = ( ) 2 + 2( )( ) + ( ) 2 = (x + 1) 2 = ( )2 1. Factorize 2. Factorize (a) x2 + 14 x + 49, (a) 4 + 4 m + m2, (b) s2 + 20 s + 100. (b) 16 + 8 x + x2. ○○○→→→ Ex 1B 1, 3–5, 11, 12 Example 2 Instant Drill 2 Factorize Factorize (a) 9x2 + 6 x + 1, (a) 4h2 + 20 h + 25, (b) 4y2 + 12 y + 9. (b) 9k2 + 24 k + 16. 9x2 = (3 x)2 Sol (a) 9x2 + 6 x + 1 Sol (a) 4h2 + 20 h + 25 = (3 x)2 + 2(3 x)(1) + 1 2 = ( ) 2 + 2( )( ) + ( ) 2 = (3 x + 1) 2 = ( )2 (b) 4y2 + 12 y + 9 (b) 9k2 + 24 k + 16 = (2 y)2 + 2(2 y)(3) + 3 2 = = (2 y + 3) 2 3. Factorize 4. Factorize (a) 81 x2 + 18 x + 1, (a) 25 y2 + 60 y + 36, Rearrange the (b) 9n2 + 42 n + 49. (b) 49 m2 + 4 + 28 m. terms. 11 ○○○→→→ Ex 1B 6, 13 (II) Square of the Difference of Two Numbers a2 – 2ab + b2 ≡ (a – b)2 Example 3 Instant Drill 3 Factorize x2 – 8x + 16. Factorize x2 – 14 x + 49. Sol x2 – 8x + 16 a = x, b = Sol x2 – 14 x + 49 a = ___, b = ___ 4 = x2 – 2( x)(4) + 4 2 = ( ) 2 – 2( )( ) + ( ) 2 = (x – 4) 2 = ( )2 5. Factorize 6. Factorize (a) x2 – 4x + 4, (a) 1 – 2y + y2, (b) s2 – 16 s + 64. (b) 36 – 12 k + k2. ○○○→→→ Ex 1B 2, 7–9 Example 4 Instant Drill 4 Factorize Factorize (a) 16 x2 – 8x + 1, (a) 25 h2 – 30 h + 9, (b) 9y2 – 12 y + 4. (b) 36k2 – 60 k + 25. Sol (a) 16 x2 – 8x + 1 Sol (a) 25 h2 – 30 h + 9 = (4 x)2 – 2(4 x)(1) + 1 2 = ( ) 2 – 2( )( ) + ( ) 2 = (4 x – 1) 2 = ( )2 (b) 9y2 – 12 y + 4 (b) 36k2 – 60 k + 25 = (3 y)2 – 2(3 y)(2) + 2 2 = = (3 y – 2) 2 7.