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Chapter 3 Factoring 1. Write the prime factorization of 630. 630 2 315 5 63 3 21 3 = 2 x 32 x 5 x 7 7 2. Determine the greatest common factor of 56 and 88. 56 1, 2, 4, 7, 8, 14, 28, 56 88 1, 2, 4, 8, 11, 22, 44, 88 2. Determine the greatest common factor of 56 and 88. 88 56 2 28 2 14 2 44 2 2 7 GCF = 2x2x2 = 8 22 2 11 3. Determine the least common multiple of 10 and 22. 10 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 22 22, 44, 66, 88, 110, 132, 154, 176, 198, 220, 240 3. Determine the least common multiple of 10 and 22. 10 2 22 5 2 =2x5 11 = 2 x 11 Circle prime factors so the highest power of each prime is selected, then multiply those to find LCM LCM = 2x5x11 LCM = 110 4. Determine the edge length of this cube. V w h V = (x)(x)(x) Volume = 91 125 cm 3 91125 = x3 3 91125 x x = 45 cm 5. Factor the binomial 44a 99a 2 = 11a(4 + 9a) 6. Factor the trinomial 24c d 40c d 32cd 3 2 2 = -8cd(3c2 + 5cd + 4d2) 3 7. Expand and simplify: 5m 3n 2 5m 3n(5m 3n) = 25m2 – 15mn – 15mn + 9n2 = 25m2 – 30mn + 9n2 8. Expand and simplify: 8h 3(7h2 4h 1) = 56h3 – 32h2 + 8h +21h2 – 12h +3 = 56h3 – 11h2 – 4h +3 9. Expand and simplify: 2x 2 5x 6 (5x 2 x 3) 2 = 10x4 – 4x3 + 6x2 + 25x3 – 10x2 + 15x – 30x2 + 12x – 18 = 10x4 +21x3 – 34x2 + 27x – 18 10. Expand and simplify: 6 x y 3x 8 y 2 x 3 y 2 6x y3x 8 y 2x 3y 2x 3y = (18x2 + 48xy – 3xy – 8y2) – (4x2 – 6xy – 6xy +9y2) = (18x2 + 45xy – 8y2) – (4x2 – 12xy +9y2) = 14x2 + 57xy – 17y2 11. Factor the following: Step 1 -12 a) x 4 x 12 2 = (x + 6)(x – 2) 2 3 Is there a common factor? No Multiply (+1)(-12) Look for numbers: ___ x ___ -12 ___ + ___ +4 +6 & -2 4 Split into Brackets. First coefficient is always the same as original expression. 5 Divide by the GCF in each bracket 11. Factor the following: +36 b) 9c 2 12c 4 = (9c – 6)(9c – 6 ) 3 3 = (3c– 2)(3c– 2) Step 1 2 3 Is there a common factor? No Multiply (+9)(+4) Look for numbers: ___ x ___ +36 ___ + ___ -12 -6 & -6 4 Split into Brackets. First coefficient is always the same as original expression. 5 Divide by the GCF in each bracket = (3c– 2)2 11. Factor the following: Step 1 c) 24b 2 50b 14 -84 = 2( 12b2 + 25b – 7) = 2(12b + 28)(12b – 3 ) 4 3 2 3 Is there a common factor? Yes Multiply (+12)(-7) Look for numbers: ___ x ___ -84 ___ + ___ +25 +28 & -3 4 Split into Brackets. First coefficient is always the same as original expression. 5 Divide by the GCF in each bracket =2(3b + 7)(4b – 1) 11. Factor the following: Difference of Squares d) 49 s 64t 2 Step 1 2 2 = (7s + 8t)(7s – 8t) 3 Is there a common factor? No Take the square root of both terms and separate into two sets of brackets. One Positive and One Negative 11. Factor the following: -40 e) 8c 2 18cd 5d 2 = (8c + 20d)(8c – 2d) 2 4 =(2c + 5d)(4c – d) Step 1 2 3 Is there a common factor? No Multiply (+8)(-5) Look for numbers: ___ x ___ -40 ___ + ___ +18 +20 & -2 4 Split into Brackets. First coefficient is always the same as original expression. 5 Divide by the GCF in each bracket 11. Factor the following: Difference of Squares f) Step 1 2 3 z 4 768 z 2 = 3z2( z2 – 256) = 3z2(z + 16)(z – 16) 3 Is there a common factor? Yes Take the square root of both terms and separate into two sets of brackets. One Positive and One Negative 12. Calculate the area of the shaded region Large Rectangle: 2x 33x 1 = 6x2 – 2x + 9x – 3 = 6x2 + 7x – 3 Small Rectangle: AShaded = (6x2 + 7x – 3) – (2x2 – x) x2 x 1 = 2x2 – x = 4x2 + 8x – 3 Algebra Tiles - 2 +x +x 2 x +x +1 -x -x -1 13. Draw the following factors using algebra tiles. There is a legend on your formula sheet: +6 a) This is the number of small tiles x 5x 6 2 (x + 2) This is the number of big tiles Step 1: Factor = (x + 3)(x + 2) (x + 3) Represents the long skinny tiles 13. Draw the following factors using algebra tiles. There is a legend on your formula sheet: This is the number of small tiles +2 b) x 3x 2 2 = (x – 2)(x – 1) This is the number of big tiles (x – 2) (x – 1) Step 1: Factor Represents the long skinny tiles 13. Draw the following factors using algebra tiles. There is a legend on your formula sheet: -12 c) This is the number of small tiles Step 1: Factor 2x x 6 __ x __ = -12 __+__ = +1 2 = (2x + 4)(2x – 3) This is the number of big tiles (x + 2) (2x – 3) 2 1 = (x + 2)(2x – 3) Represents the long skinny tiles 13. Draw the following factors using algebra tiles. There is a legend on your formula sheet: -3 d) This is the number of small tiles x 2x 3 __ x __ = -3 __+__ = -2 2 = (x + 1)(x – 3) This is the number of big tiles (x – 3) (x + 1) Step 1: Factor Represents the long skinny tiles