MHR Principles of Mathematics 10 Solutions 1


 Joel Sparks
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1 Chapter 5 Quadratic Expressions Chapter 5 Get Ready Chapter 5 Get Ready Question 1 Page 08 a 3y has one term. It is a monomial. 3 b 5+ 6a has two terms. It is a binomial. c 6 x + x 1 has three terms. It is a trinomial d 8ab 6ab + ab has three terms. It is a trinomial. 3 e 5de 7ehas two terms. It is a binomial. f 19m+ 8n 3phas three terms. It is a trinomial. Chapter 5 Get Ready Question Page 08 a y 4y + y has a degree of b 8ab + 9ab 6ab has a degree of c 10xy 3x y + 5x y has a degree of 9. d 6abc 5a bc 7abc has a degree of 5. MHR Principles of Mathematics 10 Solutions 1
2 Chapter 5 Get Ready Question 3 Page 08 5x+ 7 + x 11 = 5x+ 7+ x 11 a ( ( = 7x 4 b ( 3b 8 ( 6b 7 = ( 3b 8 + ( 6b+ 7 c ( = 3b 8 6b+ 7 = 3b 1 ( x + x+ + x + x = x + x+ + x + x = x + x d ( 9y 3 7y + 6 ( 3y 3 5y + 8 = ( 9y 3 7y ( 3y 3 + 5y 8 e ( ( = y y + y + y 3 6 = y y a + a + a a = a + a + a a = a + a f ( c 3c+ 1 ( c 3c 5 = ( c 3c+ 1 + ( c + 3c+ 5 = c c+ + c + c = c MHR Principles of Mathematics 10 Solutions
3 Chapter 5 Get Ready Question 4 Page 08 7x + 3xy y + 8x xy y = 7x + 3xy y + 8x xy y a ( ( = 15x + xy 3y b ( 4g + gh 7h ( g gh+ 3h = ( 4g + gh 7h + ( g + gh 3h c ( ( = 4g + gh 7h g + gh 3h = 3g + 3gh 10h ab + a b + ab a + b = ab + a b + ab a + b = 8ab + a + 5b d ( 3cd + c + 9d ( cd + c d = ( 3cd + c + 9d + ( cd c + d e ( ( ( = cd c d cd c d = cd + 10d x + 8 6x 7 + 5x 1 = x+ 8 6x+ 7+ 5x 1 = x + 14 f ( ( ( ( ( ( 5a b + 6b a b + 7a = 5a b + 6b a + b 7a = 5a b+ 6b a b 7a = 4a + 5b b MHR Principles of Mathematics 10 Solutions 3
4 Chapter 5 Get Ready Question 5 Page 09 a 3(x + b 4(x + c x(x + 3 d 4x(x MHR Principles of Mathematics 10 Solutions
5 Chapter 5 Get Ready Question 6 Page 09 a 7m( 3m+ 8 = 7m( 3m + 7m( 8 = + 1m 56 m b 4( c+ 9 = 4( c + ( 4( 9 = 4c 36 c 5a ( 6a 8a = 5a ( 6a + 5a ( 8a = 30a 40a 4 3 d ( d d + 1 = ( d + ( d + ( 1 = + d 4d Chapter 5 Get Ready Question 7 Page 09 a V = ( x( 6x( 5x 1 = 1x ( 5x 1 = 1x ( 5x + 1x ( 1 = 60x 1x 3 b A= ( x( 6x + ( x( 5x ( x( 5x 1 = 4x + 4x( 5x 1 + 1x( 5x 1 = 4x + 4x( 5x + 4x( 1 + 1x( 5x + 1x( 1 = 4x + 0x 4x+ 60x 1x = x x Chapter 5 Get Ready Question 8 Page 09 a The factors of 10 are 1,, 5, and 10. b The factors of 4 are 1,, 3, 4, 6, 8, 1, and 4. c The factors of 16 are 1,, 4, 8, and 16. d The factors of 3 are 1,, 4, 8, 16, and 3. Chapter 5 Get Ready Question 9 Page 09 a 8= b 14 = 7 c 8 = 7 d 30 = 3 5 MHR Principles of Mathematics 10 Solutions 5
6 Chapter 5 Get Ready Question 10 Page 09 a 6= 3 9= 3 3 The GCF of 6 and 9 is 3. b 5 = = 3 5 The GCF of 15 and 5 is 5. c 4 = 3 16= The GCF of 16 and 4 is, or 8. d 0 = 5 8 = 7 The GCF of 0 and 8 is, or 4. e 36 = = 3 5 The GCF of 36 and 15 is 3. f 3= 40 = 5 The GCF of 3 and 40 is, or 8. 6 MHR Principles of Mathematics 10 Solutions
7 Chapter 5 Section 1: Multiply Polynomials Chapter 5 Section 1 Question 1 Page 17 a The model illustrates ( ( x+ 1 x+ 3 = x + 5x+ 3 b The model illustrates ( ( x+ 1 x+ 3 = x + 4x+3 c The model illustrates ( ( x+ x+ 3 = x + 4x+4 d The model illustrates ( ( x+ 3 x+ 1 = x + 7x+ 3 MHR Principles of Mathematics 10 Solutions 7
8 Chapter 5 Section 1 Question Page 17 a ( x+ 1( x+ 1 b ( x+ 4( x+ c ( x+ 1( x+ 5 8 MHR Principles of Mathematics 10 Solutions
9 + 1 3x+ d ( x ( Chapter 5 Section 1 Question 3 Page 17 a ( x+ 3( x+ 5 = x( x ( x+ 5 b ( x+ 3( x+ 4 = x( x ( x+ 4 = x + x+ x+ = x + x = x + x+ x+ = x + x c ( y+ ( y+ 4 = y( y+ 4 + ( y+ 4 d ( r+ 4( r+ = r( r+ + 4( r+ = y + y+ y+ = y + y = r + r+ r+ = r + r e ( n+ 7( n+ 1 = n( n ( n+ 1 f ( p+ 9( p+ 9 = p( p ( p+9 = n + n+ n+ = n + n = p + p+ p+ = p + p g ( w+ 7( w+ 8 = w( w ( w+ 8 h ( d + 3( d + 11 = d( d ( d + 11 = w + w+ w+ = w + w = d + d + d + = d + d MHR Principles of Mathematics 10 Solutions 9
10 Chapter 5 Section 1 Question 4 Page 17 a ( k 3( k 5 = k( k 5 3( k 5 b ( y 3( y 4 = y( y 4 3( y 4 = k k k + = k k = y y y+ = y y c ( x ( x 4 = x( x 4 ( x 4 d ( q 4( q = q( q 4( q = x x x+ = x x = q q q+ = q q e ( j 7( j 1 = j( j 1 7( j 1 f ( p 9( p 3 = p( p 3 9( p 3 = j j j+ = j j = p p p+ = p p g ( z 7x( z 8x = z( z 8x 7x( z 8x = z 8xz 7xz+ 56x = z 15xz+ 56x h ( b 3c( b 11c = b( b 11c 3c( b 11c = b 11bc 3bc+ 33c = b 14bc+ 33c Chapter 5 Section 1 Question 5 Page 18 a ( x+ 3( x 5 = x( x 5 + 3( x 5 b ( y+ 3( y 4 = y( y 4 + 3( y 4 = x x+ x = x x = y y+ y = y y c ( c ( c+ 4 = c( c+ 4 ( c+ 4 = c + c c = c + c d ( w 4( w+ = w( w+ 4( w+ = w + w w = w w e ( m+ 7( m 1 = m( m 1 + 7( m 1 f ( y 9( y+ 3 = y( y+ 3 9( y+3 = m m+ m = m + m = y + y y = y y g ( x + 7y( x 8y = x( x 8y + 7y( x 8y = x 8xy + 7xy 56y = x xy 56y h ( a+ 6b( a 10b = a( a 10b + 6b( a 10b = a 10ab+ 6ab 60b = a 4ab 60b 10 MHR Principles of Mathematics 10 Solutions
11 Chapter 5 Section 1 Question 6 Page 18 a ( ( x+ 3 x+ 4 = x + 8x+ 3x+ 1 = + + x 11x 1 y 3 5y 7 = 5y 7y 15y+ 1 b ( ( = + 5y y 1 c ( 6c 1 ( 3c+ 5 = 18c + 30c 3c 5 d = + 18c 7c 5 ( ( 7w w+ 1 = 14w + 7w 4w = + 14w 3w e ( ( 5m+ 6 5m 6 = 5m 30m+ 30m 36 = 5m 36 9y y+ = 18y + 18y 4y 4 f ( ( = + 18y 14y 4 g ( ( 7d + 5c 8d 6c = 56d 4cd + 40cd 30c = 56d cd 30c 6q+ 5r 7q 1r = 4q 7qr+ 35qr 60r h ( ( = 4q 37qr 60r Chapter 5 Section 1 Question 7 Page 18 a 3( x 5( x+ 6 = 3( x + 6x 5x 30 b ( x 7( x 9 = ( x 9x 7x+ 63 = 3( x + x 30 = ( x 16x = x + x 3 16 = x + x c ( y+ ( y 8 = ( y 8y+ y 16 d ( k + 3( k + 7 = ( k + 7k + 3k + 1 = ( y 6y 16 = ( k + 10k + 1 = + + y 6y 16 e m( m 3n( m 5n = m( m 5mn 3mn+ 15n = m( m 8mn+ 15n = m 8m n+ 15mn 3 f a( 3a+ 4b( 6a+ 7b = a( 18a + 1ab+ 4ab+ 8b = a( 18a + 45ab+ 8b = 36a + 90a b+ 56ab 3 = + + k 0k 4 MHR Principles of Mathematics 10 Solutions 11
12 Chapter 5 Section 1 Question 8 Page 18 a ( ( ( ( x+ 4 x+ 6 + x 1 x+ 7 = x + 6x+ 4x+ 4+ x + 7x x 7 = + + x 16x 17 b ( x+ 5( 3x 7 + ( 4x+ 9( x 11 = 6x 14x+ 15x 35+ ( 8x 44x+ 18x 99 = x x+ x + x x+ x = x x c 3( 6x ( 6x 1 ( x 3( 5x+ 6 = 3( 36x 6x 1x+ ( 10x + 1x 15x 18 = 336 ( x 18x+ 110 ( x 3x 18 = x 54x 6 10x 3x 18 = x x d ( x ( x 3 + ( 3x+ 5( x+ 4 = ( x 3x x+ 6 + ( 3x + 1x+ 5x+ 0 = 1( x 5x+ 6 + ( 3x + 17x+ 0 = x + x + x + x+ = x + x ( ( ( ( ( e ( x+ 4 x 4 = x+ 4 x+ 4 x 4 x 4 ( ( = x + x+ x+ x x x = x + x+ x x = x + x+ x + x = 16x f 53 ( x 1( 5x + 66 ( x+ 3( 5x = 515 ( x 6x 5x ( x 1x+ 15x 6 = 515 ( x 11x ( x + 3x 6 75x 55x x 18x 36 = = x + x 1 MHR Principles of Mathematics 10 Solutions
13 Chapter 5 Section 1 Question 9 Page 18 a h= ( d 3( d 15 ( d d d 5 ( d d = = = d + d b h = ( d 3( d 15 h = d + 36d 90 = ( 10 3( ( 10 ( 10 = 70 = = 70 Both forms of the equation predict a height of 70 m when d represents 10 m. Chapter 5 Section 1 Question 10 Page 18 a A1 = x b A = ( x+ ( x+ 3 6 c A = ( x+ 3( x+ 6 = x + 6x+ 3x+ 18 = x + 9x+ 18 d e A A1 = x + 9x+ 18 x = 9x + 18 A A1 = 9x+ 18 ( 1 = = 16 If x represents 1 m, the increase in area is 16 m. MHR Principles of Mathematics 10 Solutions 13
14 Chapter 5 Section 1 Question 11 Page 18 a i A= x( x+ 10 = x + 10x ii A = x( x = x iii A= ( x+ 5( x+ 6 = x + x+ x+ = x + x b Chapter 5 Section 1 Question 1 Page 18 a The xintercepts of y= ( x+ 3( x 1 are 3 and 1. b y = ( x+ 3( x 1 c = x x+ x = x + x MHR Principles of Mathematics 10 Solutions
15 Chapter 5 Section 1 Question 13 Page 19 a b V = w( w+ ( c V = w( w+ ( = ww ( + = + w 4 w Chapter 5 Section 1 Question 14 Page 19 a b SA = 6( x( x 1 = 6x c SA = ( x + y 6 d SA SA1 = 6( x + y 6x ( ( = 6 x+ y x+ y 6x ( ( = 6 x + xy + xy + y 6x = 6 x + xy+ y 6x = x + xy+ y x = 1xy + 6y MHR Principles of Mathematics 10 Solutions 15
16 3 3 e V V1 = ( x+ y x ( ( ( = x+ y x+ y x+ y x ( ( ( x y( x xy y x = x+ y x + xy+ xy+ y x 3 = = x xy xy xy xy y x = 3xy+ 3xy + y 3 Chapter 5 Section 1 Question 15 Page 19 a d = 3000( v+ 0.3( 1.0 v ( v v v ( v v = = = v + v b d = v + v+ = 3000 = 100 ( 0. ( At 0. m/s, she can swim 100 m before running out of air. c She can swim a maximum of m at 0.35 m/s before running out of air. 16 MHR Principles of Mathematics 10 Solutions
17 Chapter 5 Section 1 Question 16 Page 19 a Method 1: ( ( 4 x x = 4x+ 4+ 5x = 9x + 4 Method : ( 4( 6 ( x + x+ x x+ = x + x+ x+ x x = 9x + 4 b Method 1: ( ( x x = x + 4x+ 6 Method : ( ( x x+ 7 3 x = x + 7x 3x+ 6 = x + x+ 4 6 Chapter 5 Section 1 Question 17 Page 19 a n = p n 500 = 100 p n 500 = p 100 n = p n 5 = p 100 b R = np n = n n = 5n 100 c MHR Principles of Mathematics 10 Solutions 17
18 Chapter 5 Section 1 Question 18 Page 19 s n n = + 1 Chapter 5 Section 1 Question 19 Page 19 Answers will vary. Chapter 5 Section 1 Question 0 Page 19 The number n must be 18 more than a multiple of 4. The smallest possible value for n is = 4. If 4 is divided by 8, the remainder is. Answer C 18 MHR Principles of Mathematics 10 Solutions
19 Chapter 5 Section Special Products Chapter 5 Section Question 1 Page 5 a b c d MHR Principles of Mathematics 10 Solutions 19
20 Chapter 5 Section Question Page 5 Use the appropriate pattern for squaring a binomial. a x+ 5 = x + x b ( y+ 4 = y + y ( ( ( ( ( = x + x c ( w+ 6 = ( w + ( w( 6 + ( 6 d ( = w + w e ( m+ 11 = ( m + ( m( 11 + ( 11 f ( = m + m+ 11 ( ( ( ( ( k + 7 = ( k + ( k( 7 + ( 7 c+ 10 = ( c + ( c( 10 + ( 10 g g+ 9 = g + g 9 + h ( = g + g ( ( ( ( = y + y = k + k = c + c ( ( ( ( 9 x+ 0 = x + x = x + x Chapter 5 Section Question 3 Page 5 Use the appropriate pattern for squaring a binomial. a ( x 5 = ( x ( x( 5 + ( 5 b ( = x x c ( x 9 = ( x ( x( 9 + ( 9 d ( = x x e ( v 1 = ( v ( v( 1 + ( 1 f ( = v v g ( n = ( n ( n( + ( h ( = n n+ 4 4 Chapter 5 Section Question 4 Page 5 Use the appropriate pattern for squaring a binomial. a ( x + 3y = ( x + ( x( 3y + ( 3y = x + 6xy+ 9y c ( 5c+ d = ( 5c + ( 5c( d + ( d d ( = 5c + 0cd + 4d e ( 9k + m = ( 9k + ( 9k( m + ( m f ( = 81k + 36km+ 4m 0 MHR Principles of Mathematics 10 Solutions z 3 = ( z ( z( 3 + ( 3 = z z+ 6 9 c 1 = ( c (( c 1 + ( 1 = c c+ 1 ( ( ( ( b 100 = b b = b b+ m 6 = ( m ( m( 6 + ( 6 = m m b ( x y = ( x ( x( y + ( y = 4x 4xy+ y ( ( ( ( 3a 4b = 3a 3a 4b + 4b = 9a 4ab+ 16b ( ( ( ( 4u 5v = 4u 4u 5v + 5v = 16u 40uv+ 5v
21 Chapter 5 Section Question 5 Page 5 Use the appropriate pattern for the product of a sum and a difference. a ( ( ( v+ 1 v 1 = v ( 1 b ( a 1 ( a+ 1 = ( a ( 1 = v 1 = a c ( y+ 5 ( y 5 = ( y ( 5 d ( x 7 ( x+ 7 = ( x ( 7 = y 5 e ( e 9 ( e+ 9 = ( e ( 9 f ( z+ 6 ( z 6 = ( z ( 6 = e 81 g ( x+ 1( x 1 = ( x ( 1 h ( y 3 ( y+ 3 = ( y ( 3 = x 144 Chapter 5 Section Question 6 Page 5 a ( w v( w+ v = ( w ( v b ( 3m n ( 3m+ n = ( 3m ( n c ( y+ 6x ( y 6x = ( y ( 6x d ( 3x + 4y ( 3x 4y = ( 3x ( 4y 1 = x = z Use the appropriate pattern for the product of a sum and a difference. = w v = y 36x = y 9 = 9m n = 9x 16y e ( ( + = ( ( 7g 3h 7g 3h 7g 3h = 49g 9h f ( ( + = ( ( 9x 8y 9x 8y 9x 8y = 81x 64 y MHR Principles of Mathematics 10 Solutions 1
22 Chapter 5 Section Question 7 Page 5 a ( x+ 4( x 4 = ( x ( 4 = x 16 ( x 4( x 4 ( ( + = = 1 x 16 = 16 = 1 b ( x 8 = ( x ( x( 8 + ( 8 x 16x 64 = + x ( x 8 = ( 8 = 36 ( 16x+ 64 = = 36 c ( x+ 8 = ( x + ( x( 8 + ( 8 x 16x 64 = + + x ( x + 8 = ( + 8 = 100 ( + 16x+ 64 = = 100 MHR Principles of Mathematics 10 Solutions
23 d ( x 10( x+ 10 = ( x ( 10 = x 100 ( x 10( x 10 ( 10( + = + 10 = 96 x 100 = 100 = 96 e ( x+ 11( x 11 = ( x ( 11 = x 11 ( x ( x ( ( = = 117 f x 11 = 11 = 117 ( x+ 1 = ( x + ( x( 1 + ( = x + x+ x ( x + 1 = ( + 1 = 196 ( + 4x+ 144 = = 196 g ( x 7 = ( x ( x( 7 + ( 7 = x x x ( x 7 = ( 7 = 5 ( 14x+ 49 = = 5 MHR Principles of Mathematics 10 Solutions 3
24 h ( x 30( x+ 30 = ( x ( 30 = x 900 ( x ( x ( ( = = 896 x 900 = 900 = 896 Chapter 5 Section Question 8 Page 6 ( A= π r+ k ( r ( r( k ( k = π + + ( r rk k = π + + = π r + πrk + πk Chapter 5 Section Question 9 Page 6 a b A= ( x+ 5( x+ 5 = x + x c Increase in area = x + 10x+ 5 x = 10x MHR Principles of Mathematics 10 Solutions
25 Chapter 5 Section Question 10 Page 6 a The vertex is (, 0. b y = ( x+ ( x ( x( ( = + + = x + x+ 4 4 c L.S. = y = R.S. = x + x = ( 4( = 0 L.S. = R.S The point (, 0 satisfies the equation y= x + 4x+ 4. Chapter 5 Section Question 11 Page 6 a A= ( 3x+ y( 3x y ( 3x ( y = = 9x 4y b = ( + ( ( Change in area 3x y 3x y 3x = 9x 4y 9x = 4y c A= 9x 4y ( 8 ( 5 = 9 4 = 476 Change in area = 4y = 4 ( 5 = 100 The area of the rectangle is 476 cm. The change in area is 100 cm less. MHR Principles of Mathematics 10 Solutions 5
26 Chapter 5 Section Question 1 Page 6 Method 1: A= x+ x + ( ( ( 4 ( x ( = + 8 = + x = + x Method : ( A= x + 4x ( ( ( ( = x x + + 4x x x x = x = + 4 The expressions are equivalent. Chapter 5 Section Question 13 Page 6 a b ( x ( x Exposed area = ( x ( x( 5 ( 5 ( x ( x( ( = = x + x+ x x = 1x + 1 c Exposed area = ( x+ ( x 1 ( x ( x( ( ( x ( x( 1 ( 1 = = x + x+ x + x = 1x MHR Principles of Mathematics 10 Solutions
27 Chapter 5 Section Question 14 Page 6 ( ( a 31 9 = b = = 30 1 ( ( = 60 1 = = = 899 = 3599 c = ( 100 1( = = = 9999 d = ( ( 70 1 = 70 1 = = 4899 Chapter 5 Section Question 15 Page 6 ( ( 3 8 = = 30 = = 896 ( ( a = b 35 5 = = 80 4 ( ( = 30 5 = = = 6384 = 875 c 96 = ( ( d = = = 9984 ( ( = = 80 3 = = 6391 Chapter 5 Section Question 16 Page 7 a h= 5 t b ( MHR Principles of Mathematics 10 Solutions 7
28 c h ( t = ( t (( t ( = ( t t = = + + 5t 30t = + 5t 30t 35 Chapter 5 Section Question 17 Page 7 a R= ( x( x = x + 448x+ x = x+ x b R= x+ x ( ( 000 = = The resolution of the new CCD is 14.7 megapixels. Chapter 5 Section Question 18 Page 7 Solutions for Achievement Checks are shown in the Teacher s Resource. Chapter 5 Section Question 19 Page 7 4 a ( x = ( x ( x ( x 4x 4( x 4x 4 = = x 4x + 4x 4x + 16x 16x+ 4x 16x x x x x = b ( x+ 3( x 5( 4x+ 7 = ( x+ 3( 4x + 7x 0x 35 = ( x+ 3( 4x 13x 35 = x x x+ x x = x x x MHR Principles of Mathematics 10 Solutions
29 c ( x + 5x+ 3 = ( x + 5x+ 3( x + 5x = x + x + x + x + x + x+ x + x = x + x + x + x+ 3 d ( 5x = ( 5x ( 5x ( 5x ( 5x 0x 4 = + = x 100x 0x 50x 40x 8 = x x + x Chapter 5 Section Question 0 Page a Δ E = mv m( v b Δ E = mv m( v x = mv m( v 10v = = 5mv 1.5m c Δ E = mv m( v 5 mv mv mv m 1 1 Δ E = mv m( v x 1 1 = mv m v xv + x 1 1 = = mvx 0.5mx ( mv mv mvx mx MHR Principles of Mathematics 10 Solutions 9
30 Chapter 5 Section Question 1 Page 7 a b ( + 1 ( + 1 ( +1 n n n n n n = n = ( n MHR Principles of Mathematics 10 Solutions
31 Chapter 5 Section 3 Common Factors Chapter 5 Section 3 Question 1 Page 34 a The GCF is x. b The GCF is a. c The GCF is x. d The GCF is k 4 / e The GCF is m. f The GCF is 3y. Chapter 5 Section 3 Question Page 34 a b MHR Principles of Mathematics 10 Solutions 31
32 c Chapter 5 Section 3 Question 3 Page 34 ( a 15w+ 5z = 5 3w+ 5z b 3a 11 b cannot be factored c 17ca 8cd c( 17a 8d = d 9y 8y 3 = y( 9 8y + = ( + 4g 8g + 6= ( g 4g e 1b 18b 6b b 3 f + n 5 + 1n 4 6n 3 = n 3 ( n + 6n 3 g 7h 3m 5 k cannot be factored h Chapter 5 Section 3 Question 4 Page 34 a 14x y 16xy 3 xy( 7x 8y + = + b 10km 3 6km = km ( 5k 3 ( c 8 sy+ 11 t cannot be factored d 66cde 4 cde = cde 3c 1 e 7 gh mn 13 pq cannot be factored f 5fg 5fg + 0f g = 5fg g 5+ 4f + ( g 7r s 18r 3 s 36rs 3 = 9rs ( 3r r 4s h 4n p n 4 p 1n 3 p = n p ( p+ 5n 6n 3 MHR Principles of Mathematics 10 Solutions
33 Chapter 5 Section 3 Question 5 Page 34 a 3x( x ( x+ 8 = ( x+ 8( 3x+ b c y x 5 4 x 5 cannot be factored d 5 ab ( cb ( + 1 = ( b+ 1( a+ 9c ( + ( + 4s( r+ u t( r+ u = ( r+ u( 4s t Chapter 5 Section 3 Question 6 Page 34 a mx + my + x + y = ( mx + my + ( x + y = m( x+ y + ( x+ y = ( x+ y( m+ c ay + 3ay + 4y + 1= ( ay + 3ay + ( 4y + 1 ay( y 3 4( y 3 ( y 3( ay 4 = = + + b x + 3x+ x+ 6= ( x + 3x + ( x+ 6 x( x 3 ( x 3 ( x 3( x = = + + d 6x + 9x x 3= ( 6x + 9x + ( x 3 3x( x 3 ( x 3 ( x 3( 3x 1 = + + = + e 16v 1v 1v+ 9 = ( 16v 1v + ( 1v+ 9 4v( 4v 3 3( 4v 3 ( 4v 3( 4v 3 = = ( 4v 3 = Chapter 5 Section 3 Question 7 Page 34 Answers will vary. Sample answers are shown. a 1x 18y 6( x 3 + = + y b x 3 + x + x= x( x + x+ 1 c 5y + 10y 3 = 5y ( 1+ y d ab 3 + 4ab ab 4 5 = ab 3 ( 1+ ab+ 3ab MHR Principles of Mathematics 10 Solutions 33
34 Chapter 5 Section 3 Question 8 Page 35 a b P= l+ w ( l w = + P= l+ w ( ( 9 = 15 + = 48 ( ( 15 9 ( P= l+ w = + = 4 = 48 The perimeter is 48 cm using either formula because the formulas are equivalent. Chapter 5 Section 3 Question 9 Page 35 a b SA = πr + πrh ( h = πr r+ SA = πr + πrh ( π( ( 8 = π = 66π ( ( 3( 3 8 SA = πr r + h = π + = 66π The surface area is 66π, or about 07.3 cm, using either formula because the formulas are equivalent. Chapter 5 Section 3 Question 10 Page 35 ( ( x x x + x= x + x = x( x ( 6x 9 = = x + There are four possible lengths and widths. Chapter 5 Section 3 Question 11 Page 35 a 5x( 7 y + 4( y 7 = 5x( y 7 + 4( y 7 = ( y 7( 5x+ 4 b 5y( x 1 + ( 1 x = 5y( x 1 ( x 1 = ( x 1( 5y 34 MHR Principles of Mathematics 10 Solutions
35 Chapter 5 Section 3 Question 1 Page 35 a ( ( ( SA = x 1 + x + + x + 5 b SA = ( x 1 + ( x + + ( x + 5 = 4x 4x+ 1+ 4x + 8x+ 4+ 4x + 0x+ 5 = + + 1x 4x 30 c SA = x + x ( x x = Chapter 5 Section 3 Question 13 Page 35 a A= ( 8x( 4x ( 4y( y = 3x 8y ( x y = 84 ( x y( x y = 8 + b A= πr πr ( R r = π = π( R r( R+ r Chapter 5 Section 3 Question 14 Page 35 y = x 3x ( x 3 = x The xintercepts are 0 and 1.5. Chapter 5 Section 3 Question 15 Page a x + y = ( x + 3y b a 3 ab= a( a b km c km 4 km 3 + km = ( k 3 3m+ k This can simplify the operations when the values of the variables are known because you can wait until the last step to divide. MHR Principles of Mathematics 10 Solutions 35
36 Chapter 5 Section 3 Question 16 Page 35 3a+ 8b= 1 ( a+ b = ( a+ 40b= 60 Answer C Chapter 5 Section 3 Question 17 Page 35 Let the five consecutive numbers be x, x + 1, x +, x + 3, and x + 4. ( ( ( ( x + x+ + x+ + x+ + x+ = x + x + x+ + x + x+ + x + x+ + x + x+ 16 = + + 5x 0x 30 ( x x = The sum is divisible by MHR Principles of Mathematics 10 Solutions
37 Chapter 5 Section 4 Factor Quadratic Expressions of the Form x +bx + c Chapter 5 Section 4 Question 1 Page 40 a b = ( + 3( + 1 x + 7x+ 10= ( x+ 5( x+ x x x x c d = ( + 4( + x + 4x+ 4= ( x+ ( x+ x x x x Chapter 5 Section 4 Question Page 40 a The integers are 5 and 9. b The integers are 3 and. c The integers are and 5. d The integers are 10 and. MHR Principles of Mathematics 10 Solutions 37
38 Chapter 5 Section 4 Question 3 Page = ( + ( + j + 1 j+ 7 = ( j+ 9( j+ 3 a x 7x 10 x 5 x b c k + 5k + 4= ( k + 4( k + 1 d p + 9 p+ 1 is not factorable e w 11w 5 is not factorable f + + d + 10d + 4 = ( d + 6( d + 4 Chapter 5 Section 4 Question 4 Page 40 ( ( a m 7m+ 10= m 5 m b x 5x+ 7 is not factorable c y 5y+ 4= ( y 4( y 1 d r 16r+ 64 = ( r 8 e w 9w 4 is not factorable f + q 10q+ 9 = ( q 9( q 1 Chapter 5 Section 4 Question 5 Page 41 a a 3a 10= ( a 5( a+ b s + 3s 10= ( s ( s+ 5 c d 8d 9 ( d 9( d 1 = + d e g 5g 14 ( g 7( g = + f r f + 7 f 6 is not factorable + r 6 is not factorable g x x 4 ( x 6( x 7 + = + h b b 4 is not factorable Chapter 5 Section 4 Question 6 Page 41 x + 18x+ 80 = x+ 10 x+ 8 a ( ( x + 10 = = 5 x + 8= = 3 The length is 5 cm and the width is 3 cm. x 15x+ 50 = x 5 x 10 b ( ( x 5= 15 5 = 10 x 10 = = 5 The length is 10 cm and the width is 5 cm. 38 MHR Principles of Mathematics 10 Solutions
39 Chapter 5 Section 4 Question 7 Page 41 a 3x + 1x+ 9= 3( x + 4x+ 3 = 3( x+ 1( x+ 3 c 5z + 40z+ 60= 5( z + 8z+ 1 = 5( z+ ( z+ 6 b d d + 56 = ( d 11d + 8 = ( d 4( d 7 d 4s 8s 3= 4( s s 8 ( s ( = 4 4 s+ ( e bx + 10bx 4b = b x + 10x 4 f ( ( = b x+ 1 x ( 3 x x x x x x = = x( x+ 6( x+ 1 Chapter 5 Section 4 Question 8 Page 41 Answers may vary. For example: a b = 8 or b = 7 b b = 5 or b = 4 x + 8x+ 1= x+ x+ 6 x 5x+ 4= x 1 x 4 ( ( ( ( x x x x = ( ( ( ( 4 + 4= x x x x c b = 7 or b = d b = 9 or b = 3 x 7x 8= ( x 8( x+ 1 x + 9x 10= x 1 x+ 10 x x 8= ( x 4( x+ x + 3x 10= ( x+ 5( x ( ( Chapter 5 Section 4 Question 9 Page 41 a c = 5 or c = 9 b c = or c = 6 x + 6x+ 5= ( x+ 5( x+ 5 x + 6x+ 9= ( x+ 3( x+ 3 x x = ( x ( x+ 1 x x 6= ( x 3( x + c c = 9 or c = 0 d c = 3 or c = 8 x 8x 9= ( x 9( x+ 1 x 8x 0= ( x 10( x+ x + x 3= ( x+ 3( x 1 x + x 8= ( x+ 4( x Chapter 5 Section 4 Question 10 Page 41 x+ x+ = x + x a ( ( ( ( x + y x+ 3y = x + 4xy+ 3y c ( ( ( ( x x+ = x + x x y x+ 9y = x + 7xy 18y x+ x = x x b ( ( ( ( x + 4y x 6y = x xy 4y d ( ( ( ( x x = x x x 6y x 9y = x 15xy+ 54y The coefficients of corresponding terms in the simplified forms are the same. MHR Principles of Mathematics 10 Solutions 39
40 Chapter 5 Section 4 Question 11 Page 41 a a + 11ab+ 4b = ( a+ 3b( a+ 8b b k 11km+ 18m = ( k m( k 9m c c + 4cd 1d = ( c+ 7d( c 3d d x 6xy 16y = ( x 8y( x + y Chapter 5 Section 4 Question 1 Page 41 Answers will vary. Chapter 5 Section 4 Question 13 Page 41 x 4x 1= x 6 x+ a ( ( b The xintercepts are and 6. c x =, (, 16 Chapter 5 Section 4 Question 14 Page 41 a h= t + t ( t t = ( t ( = 5 4 t+ 1 b The binomial factors can be used to find the tintercepts, one of which represents when the ball will land on the ground. 40 MHR Principles of Mathematics 10 Solutions
41 Chapter 5 Section 4 Question 15 Page 41 a The equations are alike because the coefficients are the same. They are different because the degrees of the variables are different. x 4 + 9x + 0= x + 5 x + 4 b ( ( Chapter 5 Section 4 Question 16 Page = ( + 5( + 6 x 4 7xy + 1y = ( x 3y( x 4y 4 a x x x x b c x 6 3x 3 54= ( x 3 9( x ( d ( x + ( x = ( x + ( x = 3( x 5 ( x = 3( x 7( x Chapter 5 Section 4 Question 17 Page 41 3 a ( x+ = ( x+ ( x+ ( x+ ( x ( x 4x 4 = = x + 4x + 4x+ x + 8x+ 8 3 = x + 6x + 1x+ 8 b x 3 + 9x + 7x+ 7= ( x+ 3 3 c 8a a b+ 150ab + 15b 3 = ( a+ 5b 3 MHR Principles of Mathematics 10 Solutions 41
42 Chapter 5 Section 5 Factor Quadratic Expressions of the Form ax +bx + c Chapter 5 Section 5 Question 1 Page 46 a ( ( x + 5x+ 3= x+ 1 x+ 3 b ( ( 3x 7x+ 4= x+ 1 3x+ 4 c ( ( 6x + 5x+ 1= x+ 1 3x+ 1 4 MHR Principles of Mathematics 10 Solutions
43 d ( ( 6x + 11x+ 4= x+ 1 3x+ 4 Chapter 5 Section 5 Question Page 46 a x + 7x+ 5= x + x+ 5x+ 5 b ( x x ( 5x 5 = x( x 1 5( x 1 ( x 1( x 5 = = + + c 4k + 15k + 9= 4k + 1k + 3k + 9 d ( 4k 1k ( 3k 9 = k( k 3 3( k 3 ( k 3( 4k 3 = = y + y+ = y + y+ y+ ( 6y 16y ( 3y 8 = ( ( ( 3y 8( y 1 = y 3y y+ 8 = + + 3m + 10m+ 8= 3m + 6m+ 4m+ 8 ( 3m 6m ( 4m 8 = m( m 4( m ( m ( 3m 4 = = + + e 10w + 15w+ 3 is not factorable f 1q + 17q+ 6 = 1q + 9q+ 8q+ 6 ( 1q 9q ( 8q 6 = q( 4q 3 ( 4q 3 ( 4q 3( 3q = = + + MHR Principles of Mathematics 10 Solutions 43
44 Chapter 5 Section 5 Question 3 Page 46 a 4x 11x+ 6= 4x 8x 3x+ 6 b ( 4x 8x ( 3x 6 = + + 4x( x 3( x ( x ( 4x 3 = = c 6c 3c+ 1 is not factorable d 5n 11n+ 6= 5n 6n 5n+ 6 ( 5n 6n ( 5n 6 = + + n( 5n 6 1( 5n 6 ( 5n 6( n 1 = = a a+ = a a a+ ( 6a 6a ( a 1 = + + 6a( a 1 1( a 1 ( a 1( 6a 1 = = e 9b 4b+ 7= 9b 1b 3b+ 7 f 15k 19k + 6 = 15k 10k 9k + 6 ( 9b 1b ( 3b 7 = + + 3b( 3b 7 1( 3b 7 ( 3b 7( 3b 1 = = Chapter 5 Section 5 Question 4 Page 46 ( 15k 10k ( 9k 6 = + + 5k( 3k 3( 3k ( 3k ( 5k 3 = = a 3y + 4y 7= 3y 3y+ 7y 7 b ( 3y 3y ( 7y 7 = + 3y( y 1 7( y 1 ( y 1( 3y 7 = + = + m + 3m 9= m 3m+ 6m 9 ( m 3m ( 6m 9 = + m( m 3 3( m 3 ( m 3( m 3 = + = + c 8k 6k 5= 8k 10k + 4k 5 d 1 y + y 1 = 1 y + 4 y 3y 1 ( 8k 10k ( 4k 5 = + ( ( ( 4k 5( k 1 = k 4k 5 + 4k 5 = + ( 1 y 4 y ( 3y 1 = + + 4y( 3y 1 1( 3y 1 ( 3y 1( 4y 1 = + + = + e 9x 15x 4 cannot be factored f h h = h h+ h ( 5h 15h ( h 3 = + h( h ( ( h 3( 5h 1 = h 3 = + 44 MHR Principles of Mathematics 10 Solutions
45 Chapter 5 Section 5 Question 5 Page 46 a b c d e f x + xy+ y = x + xy+ xy+ y ( ( = 3x + 6xy + xy + y 3x( x y y( x y ( x y( 3x y = = + + 6m + 13mn+ n = 6m + 1mn+ mn+ n ( ( = 6m + 1mn + mn+ n 6m( m n n( m n ( m n( 6m n = = + + p pq+ q = p pq pq+ q ( ( = p 10pq pq 5q p( p 5q q( p 5q ( p 5q( p q = = = c cd d c cd cd d ( ( = 6c 1cd + 5cd 10d 6cc ( d 5dc ( d ( c d( 6c 5d = + = x xy y = x xy+ xy y ( ( = 9x 1xy + 3xy 4y 3x( 3x 4y y( 3x 4y ( 3x 4y( 3x y = + = + 6d + de e = 6d + 4de 3de e ( ( = 6d + 4de + 3de e d( 3d e e( 3d e ( 3d e( d e = + + = + MHR Principles of Mathematics 10 Solutions 45
46 Chapter 5 Section 5 Question 6 Page 46 a b c k k + = k k = k k k ( k k ( k = ( ( ( k ( k = k k 3 1 k 3 = 3 1 p + p = p + p = p + p p ( p p ( p = p( p ( p ( p ( p = = m m = m m = m m+ m ( m m ( m m( m ( m ( m ( m = = = d 10x + 15x 10 = 5 x + 3x = x + x x 5 4 ( x x ( x = x( x ( x ( x ( x = = e 10r r+ 4 = 5r 11r+ = + 5r 10r r ( r r ( r = r( r ( r ( r ( r = 5 1 = MHR Principles of Mathematics 10 Solutions
47 f 8y y+ 1= 4y 11y+ 6 = y y y ( y y ( y = y( y ( y ( y ( y = 4 3 = 4 3 Chapter 5 Section 5 Question 7 Page 46 a b 4x + 1x+ 5= 4x + 10x+ x+ 5 ( 4x 10x ( x 5 = ( ( ( x 5( x 1 = x x+ 5 + x+ 5 = + + x x+ = x x x ( 7x 1x ( x 6 = + + 7x( x 3 ( x 3 ( x 3( 7x = = ( 4x + 1x+ 5= = 45 ( x+ 5( x+ 1 = ( ( + 5 ( ( + 1 = 45 ( 7x 3x+ 6= = 1 ( x 3( 7x = (( 3 ( 7( = 1 ( ( c 15x x 8 = 15x 1x+ 10x 8 ( 15x 1x ( 10x 8 = + ( ( ( 5x 4( 3x = 3x 5x 4 + 5x 4 = + d 8x + 14x 4= ( 4x + 7x = ( 4x + 8x + ( x x( x ( x ( x ( x = = ( 15x x 8 = 15 8 = 48 ( 5x 4( 3x+ = ( 5( 4 ( 3( + = 48 ( ( ( 8x + 14x 4= = 56 ( x+ ( x = (( + ( 4( = 56 e x x+ = x x x ( 6x 10x ( 9x 15 = + + x( 3x 5 3( 3x 5 ( 3x 5( x 3 = = ( x x+ = = 1 ( 3x 5( x 3 = ( 3( 5 ( ( 3 = 1 ( MHR Principles of Mathematics 10 Solutions 47
48 f 5x + 18x+ 9= 5x + 15x+ 3x+ 9 ( 5x 15x ( 3x 9 = x( x 3 3( x 3 ( x 3( 5x 3 = = + + ( ( 5x + 18x+ 9= = 65 ( x+ 3( 5x+ 3 = (( + 3 ( 5( + 3 = 65 The results are the same because the expressions are equivalent. Chapter 5 Section 5 Question 8 Page 46 Answers may vary. For example: a n = 17 or n = 8 x + 17n+ 16 = x+ 16 x+ 1 ( ( ( ( x n x x = b n = 8 or n = 0 3y + 8y+ 5= 3y + 5y+ 3y+ 5 ( 3y 5y ( 3y 5 = y( 3y 5 1( 3y 5 ( 3y 5( y 1 = = + + y + y+ = y + y+ y ( 3y 15y ( 5y 5 = y( y 5 5( y 5 ( y 5( 3y 5 = = + + c n = 13 or n = 3 6a + 13ab+ 7b = 6a + 6ab+ 7ab+ 7b ( ( = 6a + 6ab + 7ab+ 7b 6a( a b 7b( a b ( a b( 6a 7b = = + + a + ab+ b = a + ab+ ab+ b ( ( = 6a + 1ab + ab+ 7b 3a( a 7b b( a 7b ( a 7b( 3a b = = MHR Principles of Mathematics 10 Solutions
49 Chapter 5 Section 5 Question 9 Page 46 Answers may vary. For example: a k = 44 or k = 8 36m + 8m 44 = 4 9m + m 11 = m + m m ( m m ( m = m( m ( m ( m ( m = = m + m = m + m = m + m m ( m m ( m = m( m ( m ( m ( m = = b k = 4 or k = 60 18y 4 y+ 4 = 6 3y 7 y+ 4 = y y y ( y y ( y y( y ( y ( y ( y = = y y = y y = y y+ y = ( y y ( y = y( y ( y ( y ( y = = MHR Principles of Mathematics 10 Solutions 49
50 c k = 56 or k = p 7 pq+ 16q = 8 7 p 9 pq+ q = 8 7p 7pq pq+ q = 8 ( 7p 7pq ( pq q + + = 8 7p( p q q( p q = 8( p q( 7p q 81p 7 pq+ 16q = 81p 36 pq 36 pq+ 16q = 81p 36 pq + 36 pq + 16q ( ( 9p( 9p 4q 4q( 9p 4q ( 9p 4q( 9p 4q ( 9p 4q = = = Chapter 5 Section 5 Question 10 Page 46 If there are two integers whose product is a c and whose sum is b, then ax + bx + c can be factored over the integers. Chapter 5 Section 5 Question 11 Page 46 It is easier to factor ax + bx + c if a and c are prime numbers because there are fewer factors to check. Chapter 5 Section 5 Question 1 Page 47 a x + x = x + x x ( 6x 16x ( 3x 8 = + + ( ( ( 3x 8( x 1 = x 3x x+ 8 = + The length is 3x + 8 and the width is x 1. b P = ( 3x+ 8 + ( x 1 = ( 3( ( ( 10 1 = 114 A= x + x ( 10 ( 10 = = 7 If x = 10 cm, the perimeter is 114 cm and the area is 7 cm. 50 MHR Principles of Mathematics 10 Solutions
51 Chapter 5 Section 5 Question 13 Page 47 h = t + t ( t t ( t t t ( t t ( t = = = t( t 5 ( t 5 ( t 5( 5t = + = + The rocket hits the ground at t = 5, or 5 s. Chapter 5 Section 5 Question 14 Page 47 a r = 0.008( p 3000( p 1000 b The range must have positive values. This occurs for 1000 p c The maximum occurs for a value of p halfway between 1000 and 3000, or 000. Chapter 5 Section 5 Question 15 Page 47 R = x x = x x 360 = x + 18x 0x 360 = ( x 18x ( 0x = = x( x ( x ( x ( x There is more than one possible answer. Examples: Number sold: 0 x, price per jacket: 36 + x Number sold: 40 x, price per jacket: 18 + x Chapter 5 Section 5 Question 16 Page 47 Solutions for Achievement Checks are shown in the Teacher s Resource. MHR Principles of Mathematics 10 Solutions 51
52 Chapter 5 Section 5 Question 17 Page 47 a b c = x x x x x 4 ( 5x 15x ( 3x 9 5x ( x 3 3( x 3 ( x 3( 5x 3 = = = + + 7x 13x y + 6y = 7x 7x y 6x y + 6y ( 7 7 ( 6 6 7x ( x y 6y ( x y ( x y ( 7x 6y = x xy + xy + y 4 4 = = 6x + 13x y 8y = 6x + 16x y 3x y 8y ( ( ( ( 3 ( 3x 3 3 8y ( x 3 y = x + xy + xy y = x 3x + 8y y 3x + 8y = + 6 d 10m 7m n 1n = 10m 15m n + 8m n 1n ( ( ( ( 3 ( m 3 3n ( 5m 4n = m m n + m n n = 5m m 3n + 4n m 3n = + Chapter 5 Section 5 Question 18 Page 47 a ( x a ( x a ( x a ( x a ( x a = = x+ a + x+ a + x+ a + 1 = x+ a x+ a x+ a + 1 ( ( ( ( ( ( ( x a 1 ( x a 1 ( x a 1( x a 1 = = b ( x b ( x b ( x b ( x b ( x b = ( x b ( x b ( x b = ( x b ( x b 1 ( x b = ( x b ( x b ( x b 1( x b = = MHR Principles of Mathematics 10 Solutions
53 Chapter 5 Section 5 Question 19 Page 47 a = x x x x x ( 8x 14x ( 4x 7 = + + ( ( ( 4x 7( x 1 = x 4x x+ 7 = + Answers may vary. The shape could be a rectangle with dimensions (x 1 and (4x + 7, a parallelogram with base (x 1 and height (4x + 7, or a triangle with base (4x and height (4x + 7, or base (x 1 and height (8x b 4x 1x y+ 9xy = x( 4x 1xy+ 9y = x( 4x 6xy 6xy+ 9y = ( + ( + x 4x 6xy 6xy 9y = x x x 3y 3y x 3y = x x y x y ( ( ( 3 ( 3 ( 3 = x x y The shape is a squarebased prism with side length (x 3y and height x. MHR Principles of Mathematics 10 Solutions 53
54 Chapter 5 Section 6 Factor a Perfect Square Trinomial and a Difference of Squares Chapter 5 Section 6 Question 1 Page 53 Use the pattern for a difference of squares. ( ( ( a x 16 = x 4 b = x+ 4 x 4 y ( y ( y ( 100 = 10 = + 10 y 10 c 9k 36= 9( k 4 ( k = 9 = 9( k + ( k d 4a 11= ( a 11 ( a ( = + 11 a 11 ( w ( w ( e f 36w 49 = 6 7 = w 7 ( ( ( ( p ( p ( 144 p 1 = 1 1 = p 1 ( g 16n 5 = 4n 5 h 100g 81 = 10g 9 = 4n+ 5 4n 5 = ( 10g+ 9( 10g 9 Chapter 5 Section 6 Question Page 53 Use the pattern for a difference of squares. ( ( h 5d = ( h ( 5d = ( m+ 7n( m 7n = ( h+ 5d( h 5d a m 49n = m 7n b ( ( c( c 100 9c = 10 3c d = c 169a 49b = ( 13a ( 7b = ( 13a+ 7b( 13a 7b e 5x 36y = ( 5x ( 6y ( ( g 16 8s = ( 81 s = 5x + 6y 5x 6y ( s = 9 = 9+ 9 ( s( s f 16c 9d = ( 4c ( 3d ( ( = 4c+ 3d 4c 3d h 75h 7g = 3( 5h 9g ( h ( g = = 35h+ 3g 5h 3g ( ( 54 MHR Principles of Mathematics 10 Solutions
55 Chapter 5 Section 6 Question 3 Page 53 Use the appropriate perfect square trinomial pattern. ( ( ( ( a x + 1x+ 36 = x + x b k + 18k+ 81 = k + k ( x 6 = + ( ( ( ( ( k 9 = + ( ( ( ( c y 6y+ 9= y y d ( y 3 = ( ( ( ( e x + 0x+ 100 = x + x f ( x 10 = + ( ( ( ( m 14m+ 49 = m m ( m 7 = ( ( ( ( ( 8 r 64 16r+ r = 8 8 r + r = Chapter 5 Section 6 Question 4 Page 53 Use the appropriate perfect square trinomial pattern. ( ( ( ( ( c 3 a 4c + 1c+ 9= c + c b 16k 8k + 1 = 4k 4k = + ( ( ( ( ( 4k 1 = ( ( ( ( c 5x + 70x+ 49 = 5x + 5x d ( 5x 7 = + e 100c 180c+ 81 = 10c 10c f ( ( ( ( ( 10c 9 = ( ( ( ( 9y 30y+ 5= 3y 3y ( 3y 5 = ( ( ( ( y+ 64y = y + 8y ( 5 8y = + Chapter 5 Section 6 Question 5 Page 54 a y is not squared. b 107 ( 6( 9 c 10 is not a perfect square. d The expression is a sum of squares, not a difference of squares. MHR Principles of Mathematics 10 Solutions 55
56 Chapter 5 Section 6 Question 6 Page 54 a 4x + 8xy + 49y = ( x + ( x( 7y + ( 7y ( x 7y = + b 9k 4k + 16m = ( 3k ( 3k( 4m + ( 4m ( 3k 4m = c d 5 p + 60 pq+ 144 q cannot be factored 9y 7 x cannot be factored e a 8ab+ 98b = ( a 14ab+ 49b ( a ( a( b ( b = ( a b = 7 f 196n 144m = 4( 49n 36m ( n ( m = = 47 ( n 6m( 7n+ 6m g 5x + 70xy+ 14 y cannot be factored h 100 f 10 fg+ 36g = 4( 5 f 30 fg+ 9g ( f ( f ( g ( g = ( f g = i 400 p 900 pq = 100 p( 4 p 9q ( ( = 100 p p 3q = 100 p( p+ 3q( p 3q 56 MHR Principles of Mathematics 10 Solutions
57 Chapter 5 Section 6 Question 7 Page 54 a ( ( A= x+ 5 x 3 b A= ( x+ 5 ( x 3 = 4x + 0x+ 5 x + 6x 9 = x + x = x + x+ x ( 3x 4x ( x 16 = x( x 8 ( x 8 ( x 8( 3x = = + + Chapter 5 Section 6 Question 8 Page 54 a b = or b = b b = 0 or b = 0 y + y+ 11 = ( y+ 11 4x + 0x+ 5= x+ 5 y y y ( + 11 = 11 c b = 4 or b = 4 d b = 5 ( = n np p n p ( = 3 7 n np p n p ( ( 4x 0x+ 5= x 5 w w w ( = + 5 e b = 5 f b = 11 or b = 11 81m 90m+ 5 = ( 9m 5 16 x 88xy+ 11y = ( 4x 11y = ( 4x + ( 11y Chapter 5 Section 6 Question 9 Page 54 Answers may vary. For example: a k = 4 or k = 9 m 4n = ( m+ n( m n m 9n = ( m+ 3n( m 3n b k = 1 or k = 5 x 9= x+ 3 x 3 ( ( ( ( 5x 9 = 5x+ 3 5x 3 c k = 16 or k = 5 49c 16 = 7c+ 4 7c 4 ( ( ( ( 49c 5 = 7c+ 5 7c 5 MHR Principles of Mathematics 10 Solutions 57
58 Chapter 5 Section 6 Question 10 Page 54 a 9a b 4abcd + 16c d = ( 3ab ( 3ab( 4cd + ( 4cd ( 3ab 4cd = b 5 ( x+ 5 = 15 ( x+ 5 = 15 + ( x ( x+ 5 = ( 0 + x( 10 x 3c+ 3c = 3c+ + 3c 3c+ 3c = 6c 4 c ( ( ( ( ( ( ( ( = 4c d 4x + 6x+ 9 cannot be factored Chapter 5 Section 6 Question 11 Page 54 ( 9x + 30x+ 5= 3x+ 5 This is a perfect square. The figure could be a square, or a parallelogram whose base is equal to its height. Chapter 5 Section 6 Question 1 Page 54 3 a x x + x = x( x x+ 1 ( ( ( 1 ( 1 = x x x + = x x ( 1 The height is x. The length and width are both x 1. b The box is a squarebased rectangular prism, so the top and bottom are squares with area (x 1 and the four sides are rectangles with area x(x MHR Principles of Mathematics 10 Solutions
59 Chapter 5 Section 6 Question 13 Page 54 a Middle: ( x ( + x 1 Bottom: ( x ( + 5 x+ b Middle: ( ( x + x 1 = 4x + 8x+ 4 4x + 4x 1 Bottom: ( ( = 1x + 3 ( x = x+ 5 x+ = 4x + 0x+ 5 4x 8x 4 c Middle: 34 ( x + 1 = 34 ( ( = 63 Bottom: 34 ( x + 7 = 34 ( (5 + 7 = 1x + 1 ( x = = 81 The exposed area of the middle layer is 63 cm. The exposed area of the bottom layer is 81 cm. Chapter 5 Section 6 Question 14 Page 54 a A= r r+ π 14π 49π ( r r = π ( r = π 7 The decrease in radius was 7 cm. πr πr 14πr+ 49π = πr πr + 14πr 49π b ( = 14πr 49π The decrease in area was (14 πr 49π cm. Chapter 5 Section 6 Question 15 Page 55 x 1= ( x+ 1( x 1 ( x 1 = ( x 1( x 1 The two expressions are not the same. MHR Principles of Mathematics 10 Solutions 59
60 Chapter 5 Section 6 Question 16 Page 55 y x x = ( x = The coordinates of the vertex are (, 0. Chapter 5 Section 6 Question 17 Page 55 a = ( ( = 6( 4 = 104 c = ( ( = 195( 1 = 195 b 37 7 = ( ( 37 7 = 64( 10 = 640 d 8 = ( 8 + ( 8 = 50( 6 = 300 Chapter 5 Section 6 Question 18 Page 55 = a s ( n b s ( n = + + = n 6n 9 = + + n 6n 5 ( n 5( n 1 = + + c s ( n = ( 10 = = n s ( 5( 1 ( 10 ( 0 s = n+ n+ = = 165 There are 165 shaded squares on the 10th diagram. d Answers will vary. Chapter 5 Section 6 Question 19 Page 55 3 V = 4πx + 0πx + 5πx = πx 4x + 0x+ 5 ( ( x = πx + 5 The figure could be a squarebased prism with height πx and base side length x + 5 or a cylinder with radius x + 5 and height x. 60 MHR Principles of Mathematics 10 Solutions
61 Chapter 5 Section 6 Question 0 Page 55 a ( x 4 16= ( x ( x 4 4 b ( x ( x ( x = ( x( x 8 = ( x = c ( 5 3 ( x y = x + y x y d k 8k + 16= ( k 4 = ( k + ( k e a 6 0a ( a = + f 4 4 y x y x y x = y x y x y x = Chapter 5 Section 6 Question 1 Page 55 a k = 7 or k = 7 ( ( 4 81x + 7x + 16 = 9x x 7x + 16 = 9x 4 = ( 3x+ ( 3x b k = 0 or k = y + 0y z + 5z = y + 5z ( ( = 5 y y z z y z MHR Principles of Mathematics 10 Solutions 61
62 Chapter 5 Section 6 Question Page 55 a b Answers may vary. For example: x 1 is one of the factors of each of the expressions and the number of terms in the other factor is equal to the degree of the original expression. The terms in the other factor form a sum where the coefficient of each of the terms is one and the terms are the sum of the descending degrees of the variable starting with 1 less than the original expression. The factored form of x 4 1 does not appear to follow the pattern. When expanded, the last two terms of this factored form result in the expression x 3 + x + x + 1, which does follow the pattern c The CAS verifies that x 6 1 follows the pattern. Chapter 5 Section 6 Question 3 Page 55 3 x x x 4 x x 4x x 4x + + = a ( ( 3 = x b m 3 64 = ( m 4( m + 4m+ 16 c 3 6 ( ( 4 7 y 15z = 3y 5z 9y + 15yz + 5z Chapter 5 Section 6 Question 4 Page 55 a ( ( 8 a+ a a+ = a a + a+ a a = a b 6 3 ( ( 4 k + 16e = k + 6e k 6k e+ 36e c 343q r 4 = ( 7q 4 + 9r 8 ( 49q 8 63q 4 r r 16 6 MHR Principles of Mathematics 10 Solutions
63 Chapter 5 Section 6 Question 5 Page 55 4 a ( a+ b = ( a+ b ( a+ b ( a ab b ( a ab b = = a a b a b a b a b ab a b ab b = a + 4a b+ 6a b + 4ab + b b 81x 4 16x 3 y + 16x y 96xy y 4 = ( 3x y 4 Chapter 5 Section 6 Question 6 Page 55 ( a b = a ab+ b = 9 Answer D = + a b ab ( = 15 3 MHR Principles of Mathematics 10 Solutions 63
64 Chapter 5 Review Chapter 5 Review Question 1 Page 56 ( ( a x+ 5 x+ 8 = x + 8x+ 5x+ 4 b = x + x x + 3y x 6y = x 6xy + 3xy 18y c ( ( = x 3xy 18y d ( ( 5a 6b a+ 9b = 10a + 45ab 1ab 54b = 10a + 33ab 54b 0 ( ( Chapter 5 Review Question Page 56 a ( k + ( k 7 = ( k 7k + k 14 = ( k 5k 14 = k + k b w( w 7v( w 3v = w( w 3wv 7wv+ 1v = w( w 10wv+ 1v = w 10w v+ 1wv 3 c 7x ( 7x + 3y( 3x + 9 y = 7x ( 1x + 63xy + 9xy + 7 y = 7x( 1x + 7xy+ 7y = 147x + 504x y+ 189xy 3 y+ 1 y+ 8 + y 8 y+ 1 = y + 8y+ y+ 8+ y + y 8y 8 d ( ( ( ( = + y y x 1 x 4 = x 4x x+4 = x x+ 5 4 e 4 x 6 4x x+ 7 11x 4 = 4 8x 10x 4x x 44x+ 77x 8 ( ( ( ( ( ( = 4( 8x 34x ( 11x + 33x = x + x + x + x = x + x MHR Principles of Mathematics 10 Solutions
65 Chapter 5 Review Question 3 Page 56 ( 9 7( A = x x+ + x = x x x = x + 16x Chapter 5 Review Question 4 Page 56 a b Chapter 5 Review Question 5 Page 56 Use the appropriate pattern for squaring a binomial. a ( ( x+ 6 = x + ( x( 6 + ( 6 b ( k + 8 = k + k = x + x ( ( ( ( c ( p+ 7 = p + p d ( = p + p ( ( ( ( = k + k r = ( r ( r( + ( = r r+ 4 4 ( ( ( ( e ( e 9 = e e f ( = e e ( ( ( ( q 0 = q q = q q MHR Principles of Mathematics 10 Solutions 65
66 Chapter 5 Review Question 6 Page 56 Use the appropriate pattern for the product of a sum and a difference. a ( b+ 6( b 6 = ( b ( 6 b ( a 7 ( a+ 7 = ( a ( 7 = b 36 c ( y+ 1 ( y 1 = ( y ( 1 d ( x 15 ( x+ 15 = ( x ( 15 = y 144 e ( e 10 ( e+ 10 = ( e ( 10 f ( x+ 6( x 6 = ( x ( 6 = e 100 Chapter 5 Review Question 7 Page 56 Use the appropriate pattern for squaring a binomial. ( ( ( ( ( a y+ 4x = y + y 4x + 4x b ( = y + 8xy+ 16x = x = a = x 5 ( ( ( ( 7m n = 7m 7m n + n = 49m 14mn+ n ( ( ( ( c ( c+ 9d = c + c 9d + 9d d = 4c + 36cd + 81d ( ( ( ( 5x+ 7y 5x 7y = 5x 7y ( x y = 5 49 = 50x + 98y e ( 3a 8c ( 3a+ 8c = ( 3a ( 8c f ( 5x 8y( 5x+ 8y = ( 5x ( 8y = 9a 64c ( 5x 64 y = = 5x + 64 y 66 MHR Principles of Mathematics 10 Solutions
67 Chapter 5 Review Question 8 Page 56 a x + 9x= x( x+ 9 b 3x + x = x( 3x+1 Chapter 5 Review Question 9 Page 56 + = ( + 11xy 9xz = x( 11y 9z a 1y 4z 1 y z b c c 3 c c( c 3 d + = + 4k 8 k = 4k( 1 k Chapter 5 Review Question 10 Page 56 a 4m + 1m+ 3m+ 9= ( 4m + 1m + ( 3m+ 9 4m( m 3 3( m 3 ( m 3( 4m 3 = = + + b 9k 6k + 6k 4= ( 9k 6k + ( 6k 4 3k( 3k ( 3k ( 3k ( 3k = + = + c 8x + 16x 5x 10= ( 8x + 16x + ( 5x 10 8x( x 5( x ( x ( 8x 5 = + + = + MHR Principles of Mathematics 10 Solutions 67
68 d 16x 1xy 1xy + 9 y = ( 16x 1xy + ( 1xy + 9 y 4x( 4x 3y 3y( 4x 3y ( 4x 3y( 4x 3y ( 4x 3y = = = Chapter 5 Review Question 11 Page 56 + = ( + ( a 8m 16m 4 4 m 4m 1 b 18c 3 + 4c 8c+ 6 = 9c 3 + 1c 4c+ 3 c ( 15m mn+ 66mn = m 15m n+ 66n d ax z az + axz = az ( x z + x Chapter 5 Review Question 1 Page 56 Length 10x + 5x, width 1. Length x + x, width 5. Length 10x + 5, width x. Length x + 1, width 5x. 68 MHR Principles of Mathematics 10 Solutions
69 Chapter 5 Review Question 13 Page 56 x + 9x+ 14= x+ x+ 7 a ( ( x + 11x+ 18 = x+ x+ 9 b ( ( c x + 4x+ 4= ( x+ ( x+ MHR Principles of Mathematics 10 Solutions 69
70 Chapter 5 Review Question 14 Page 56 + = ( ( x 8x+ 15= ( x 3( x 5 a c 17c 7 c 9 c 8 b + = ( ( x + 3x 10= ( x+ 5( x c z 14z 33 z 11 z 3 d e x + 8x 9= ( x+ 9( x 1 f x x 8= ( x 4( x+ Chapter 5 Review Question 15 Page 57 a y x x = 3 18 = ( x 6( x+ 3 b The xintercepts are 3 and 6. c 3+ 6 = 1.5 The axis of symmetry is x = 1.5. y = x 3x 18 = 1.5 ( = 0.5 The vertex is (1.5, MHR Principles of Mathematics 10 Solutions
71 Chapter 5 Review Question 16 Page 57 a x + 7x+ 6= x + 4x+ 3x+ 6 b ( x 4x ( 3x 6 = x( x 3( x ( x ( x 3 = = + + c 6a 3a+ 15= 6a 18a 5a+ 15 d ( 6a 18a ( 5a 15 = + + 6a( a 3 5( a 3 ( a 3( 6a 5 = = y + y+ = y + y+ y+ ( 6y 7y ( y 9 = y( y 9 1( y 9 ( y 9( 3y 1 = = b b+ = b b b+ ( 4b 4b ( b 1 = + + 4bb ( 1 ( b 1 ( b 1( 4b 1 = = e 1m + 0m 8 = 4( 3m + 5m = m + m m ( m m ( m = m( m ( m ( m ( m = = f 14k 31k 10 = 14k 35k + 4k 10 ( 14k 35k ( 4k 10 = + 7k( k 5 ( k 5 ( k 5( 7k = + = + Chapter 5 Review Question 17 Page 57 a b 8x + xy 15y = 8x + 1xy 10xy 15y ( ( = 8x + 1xy + 10xy 15y 4x( x 3y 5y( x 3y ( x 3y( 4x 5y = + + = + 5c 4cd + 9 d is not factorable c 1m + 13mn+ 3n = 1m + 9mn+ 4mn+ 3n ( ( = 1m + 9mn + 4mn+ 3n 3m( 4m 3n n( 4m 3n ( 4m 3n( 3m n = = + + d w 9wx 8 x is not factorable MHR Principles of Mathematics 10 Solutions 71
72 e 4x + 6xy 9 y = 3( 8x + xy 3y = 3( 8x + 6xy 4xy 3y = ( + + ( 3 8x 6xy 4xy 3y ( ( ( x y( x y = 3 x 4x+ 3y y 4x+ 3y = f 9g + 4g 10 is not factorable Chapter 5 Review Question 18 Page 57 a 8 15 A= x + x = x + x x ( 8x 1x ( 10x 15 = + + 4x( x 3 5( x 3 ( x 3( 4x 5 = + + = + The length is 4x 5, and the width is x + 3. b P = ( x+ 3 + ( 4x 5 = ( ( ( 4( 1 5 = 140 A= x + x 8 15 ( 1 ( = = 1161 The perimeter is 140 cm and the area is 1161 cm. Chapter 5 Review Question 19 Page 57 ( ( ( a x 49 = x 7 b = x+ 7 x 7 y ( y ( y ( 11 = 11 = + 11 y 11 ( ( k ( c 4k 9= k 3 d a = a = + 3 k 3 e 9w 5x = 3w 5x f ( ( ( ( = 3w+ 5x 3w 5x ( ( a = = 16( 1+ 3a( 1 3a 1 81p = 1 ( 9p = ( 1+ 9p( 1 9p 7 MHR Principles of Mathematics 10 Solutions
73 Chapter 5 Review Question 0 Page 57 a x + 14x+ 49 = ( x + ( x( 7 + ( 7 ( x 7 = + b k + 8k + 16= ( k + ( k( 4 + ( 4 ( k 4 = + c y 10y+ 5 = ( y ( y( 5 + ( 5 ( y 5 = d 5y 70y+ 49 = ( 5y ( 5y( 7 + ( 7 ( 5y 7 = e 36c 108c+ 81 = 9( 4c 1c+ 9 ( c ( c( ( = ( c = 9 3 f y+ 64y = ( 11 + ( 11( 8y + ( 8y ( 11 8y = + Chapter 5 Review Question 1 Page 57 a k = 10 or k = 10 ( 5y + 10y+ 144 = 5y+ 1 ( 5y 10y+ 144 = 5y 1 b k = 49 ( 9x 4x+ 49= 3x 7 MHR Principles of Mathematics 10 Solutions 73
74 Chapter 5 Review Question Page 57 a 9w x 4wxyz + 16y z = ( 3wx ( 3wx( 4yz + ( 4yz ( 3wx 4yz = b 65 ( y 5 = 5 ( y 5 = 5 + ( y 5 5 ( y 5 = ( 0 + y( 30 y 4 4 c 144a 499 b is not factorable d 5x + 30x+ 4 is not factorable 74 MHR Principles of Mathematics 10 Solutions
75 Chapter 5 Chapter Test Chapter 5 Chapter Test Question 1 Page 58 x+ 3 x+ 3 = x + 9x+ 9 a ( ( x+ 3 x+ 4 = x + 7x+ 1 b ( ( Chapter 5 Chapter Test Question Page 58 4x 3x 5y 8z 1x 0x y 3x 3 + = + z a ( m m m+ m m + = m m + m m + m b ( ( = m 7m 1m 9 MHR Principles of Mathematics 10 Solutions 75
76 Chapter 5 Chapter Test Question 3 Page 58 y+ 5 y+ 9 = y + 9y+ 5y+ 4 a ( ( = y + y b ( ( 4x 7 3x+ = 1x + 8x 1x 14 = c ( 6k + 1 ( 6k 1 = ( 6k ( 1 d ( w 8 = ( w ( w( 8 + ( 8 = 36k 1 ( ( ( ( e ( 4c+ 5d = 4c + 4c 5d + 5d = 16c + 40cd + 5d = w w f ( x 4( x 7 5( 8x 9( 8x+ 9 = ( x 7x 4x+ 8 5( 64x 81 = ( x 11x+ 8 5( 64x = x x+ x = x x+ Chapter 5 Chapter Test Question 4 Page 58 a d = 0.006( s+ 1 ( s s = = s + s x 13x 14 b d = 0.006( s+ 1 ( 60 = =.36 d = s + s ( ( = =.36 Both forms give the same result, a stopping distance of.36 m. Chapter 5 Chapter Test Question 5 Page 58 a 9de + 6de 3 = 3de ( 3e+ d b 15 pqr 3 5pqr pqr= 5 pqr( 3pr 5pq + 1 c 5( x+ 6 ( x+ 6 = ( x+ 6( 3 d 16x + 8x 6x 3 = ( 16x + 8x + ( 6x 3 ( ( ( x 1( 8x 3 = 8x x+ 1 3 x+ 1 = + 76 MHR Principles of Mathematics 10 Solutions
77 Chapter 5 Chapter Test Question 6 Page 58 a SA = 4x( 3x ( 3x + 4 b SA = 4x( 3x ( 3x + 4 ( = x + x+ x + x = x + x+ x + x = x + x+ c SA = x + x ( x x ( x x x ( x x ( x = = = x( x ( x ( x ( x = = Chapter 5 Chapter Test Question 7 Page 58 a x + 11x+ 4 = ( x+ 3( x+ 8 b y 15y+ 56 = ( y 7( y 8 ( ( c n n 90 = n 10 n+ 9 d x 14x + 49 = x x ( h ( h ( e f d d h 100 = 10 = + 10 h 10 ( ( ( ( ( = x = d + d = d + 8 ( ( ( ( ( MHR Principles of Mathematics 10 Solutions 77
78 Chapter 5 Chapter Test Question 8 Page 58 a 3k + 1km 36m = 3( k + 4km 1m = 3 k + 6km km 1m ( ( = 3 k + 6km + km 1m k( k m m( k m ( k m( k m = = b y + y+ = y + y+ y+ ( 8y 16y ( 3y 6 = y( y 3( y ( y ( 8y 3 = = + + c 9w 4w+ 7= 9w 1w 3w+ 7 d ( 9w 1w ( 3w 7 = + + 3w( 3w 7 ( 3w 7 ( 3w 7( 3w 1 = = ( ( ( ( 5a + 60a+ 36 = 5a + 5a ( 5a 6 = + e 11w 144 = ( 11w 1 ( w ( = w 1 f 10x 7xy 6y = 10x 1xy+ 5xy 6y ( ( = 10x 1xy + 5xy 6y x( 5x 6y y( 5x 6y ( 5x 6y( x y = + = + Chapter 5 Chapter Test Question 9 Page 58 If there are two integers whose product is 9 18 and whose sum is 10, then 9x factored over the integers. 10x + 18 can be Chapter 5 Chapter Test Question 10 Page 58 x + 13x 30 = x+ 15 x a ( ( The length is x + 15, and the width is x. b The smallest integer value of x for which this area expression makes sense is x = MHR Principles of Mathematics 10 Solutions
79 Chapter 5 Chapter Test Question 11 Page 59 a k = 13 or k = 13 ( 36x + 13x+ 11 = 6x+ 11 ( 36x 13x+ 11 = 6x 11 b k = 16 ( 49d 56d + 16 = 7d 4 c k = 36 ( = 5 6 x xy y x y d k = 5 ( = a ab b a b Chapter 5 Chapter Test Question 1 Page 59 a ( 9 ( 5( A= x+ x+ x 5 b ( 9 ( 5( A= x+ x+ x = ( x 5 x 18x 81 x 5 = 18x = c ( ( ( A= x+ 9 x+ 5 x 5 ( ( ( = = 3 ( x ( 9( 53 A= = 7 + = 3 The result is the same, 3 square units, for both expressions because the expressions are equivalent. MHR Principles of Mathematics 10 Solutions 79
80 Chapter 5 Chapter Test Question 13 Page 59 = + 6 a y ( x ( x x = = x + x = x + x+ b y = x + 4x+ 70 = ( x + 1x+ 35 ( x ( x = c The three equations represent the same parabola. They result in the same graph. Chapter 5 Chapter Test Question 14 Page 59 3 a 9x 30x + 5x= x( 9x 30x+ 5 ( 3x 5 = x The height is x. The side length of the square base is 3x 5. b The prism is a squarebased prism. The top and bottom are squares with side length 3x 5 and area (3x 5. The four sides are rectangles with width 3x 5, height x, and area x(3x 5. Chapter 5 Chapter Test Question 15 Page 59 Answers may vary. For example: a k = 4 or k = 9 4m 5= ( m+ 5( m 5 9m 5= 3m+ 5 3m 5 ( ( b k = 1 or k = 5 16d 1 = 4d + 1 4d 1 ( ( ( ( c k = 9 or k = 16 a 9b = ( a+ 3b( a 3b a 16b = a+ 4b a 4b 16d 5 = 4d + 5 4d 5 ( ( 80 MHR Principles of Mathematics 10 Solutions
81 Chapter 5 Chapter Test Question 16 Page 59 a = ( ( = 195 b = ( ( = 53 c 5 48 = ( ( 5 48 = 400 Chapter 5 Chapter Test Question 17 Page 59 a The total number of squares, s, in the nth diagram is s = 4n. b The number of shaded squares, S, in the nth diagram is S = n+ 3. c The number of unshaded squares, u, in the nth diagram is u = n n 4 3 d e u = n n 4 3 4n 4n 3n 3 = + ( 4n 4n ( 3n 3 = + 4n( n 1 3( n 1 ( n 1( 4n 3 = + = + u = n n 4 3 ( 15 ( 15 = 4 3 = 88 ( 1( 4 3 (( 15 1 ( 4( 15 3 u = n n+ = + = 88 Both forms of the formula give the same results for the 15th diagram. Chapter 5 Chapter Test Question 18 Page 59 Solutions for Achievement Checks are shown in the Teacher s Resource. MHR Principles of Mathematics 10 Solutions 81
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