Exponents and Polynomials

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1 CHAPTER 6 Exponents and Polynomials Solutions Key are you ready?. F 2. B. C 4. D 5. E ( -0) 4 9. x 0. k = = = -(2 2) = = = 2 7. ( -) 6 = (-)(-)(-)(-)(-)(-) = 4. 5 = = = = , p p = p + 9p = p 22. 8y - 4x + 2y + 7x - x = 8y + 2y - 4x + 7x - x = 0y + 2x 2. (2 + w - 5) + 6w - - 5w = w + 6w - 5w = 4 + 4w 24. 6n n = 6n + 5n - 4 = n no 26. yes; ÇÇ 8 = Ç 9 2 = yes; ÇÇ 6 = Ç 6 2 = no 29. yes: ÇÇ 00 = ÇÇ 0 2 = 0 0. yes; Ç 4 = Ç 2 2 = 2. yes; Ç = Ç 2 = 2. no 6- integer exponents Check it out!. 5 - = 5 = = m is equal to 25 m. 2a. 0-4 = = = 0,000 _ b. ( -2) -4 = = ( -2) 4 (-2)(-2)(-2)(-2) = 6 _ c. ( -2) -5 = = ( -2) 5 (-2)(-2)(-2)(-2)(-2) = - 2 d = a. p - = 4-4 = = 64 = 4a. 2r 0 m - = 2 r 0 m - = 2 m = 2 m c. g 4 = g 4 h -6 h -6 = g 4 h 6 = g 4 h 6 = = - 2 think and discuss. -2; 0; t 2. For a negative exponent in the numerator, move the power to the denominator and change the negative exponent to a positive exponent; possible answer: 2 - =. Exercises guided practice Simplifying Expressions with Negative Exponents b. 8a -2 b 0 = 8(-2) -2 (6) 0 = 8 _ ( -2) 2 = 8 (-2)(-2) = 8 4 = 2 b. r - 7 = r - = 7 r 7 = 7r For a negative exponent in the denominator, move the power to the numerator and change the negative exponent to a positive exponent; possible answer: = 4 x 5. x _. 0-7 = = = 0,000,000 m 0-7 m is equal to 0,000,000 m = 6 2 = 6 6 = 6. 0 = = - = = = 5 5 = - 25 = Holt McDougal Algebra

2 6. -8 = = = - = = - 52 = = = = (4.2) 0 = _ 0. ( -) - = = ( -) (-)(-)(-) = = b -2 = (-) -2 = _ ( -) 2 = (-)(-) = 9 4. ( m - 4) -5 = (6-4) -5 = 2-5 = = 4 4 = = = m 0 = 4 m 0 = 4 = = 7 r -7 r -7 = 7 r 7 = 7r x 0 y -4 = 2 x 0 y -4 = 2 y 4 = 2 y c 4 = c 4 d - d - = c 4 d = c 4 d. (2t) -4 = (2(2)) -4 = = = 256 = 5. 2x 0 y - = 2(7) 0 ( -4) - = 2 _ ( -4) = 2 (-4)(-4)(-4) = 2-64 = k -4 = k -4 = k 4 = k 4 9. x 0 = x 0 d - d - = x 0 d = x 0 d 2. f -4 = f -4 g -6 g -6 = g 6 f 4 = g 6 f 4 2. p 7 q - = p 7 q - = p 7 q = p 7 q practice and problem solving = = = 2 - oz is equal to 2 oz = = 4 = = 625 = = = - = = = - = = = = = 49. ( = = 2 = 69. ( -) - = _ = ( -) (-) = - 4. ( -4) 2 = (-4)(-4) = 6 5. ( 2) -2 = ( _ = 2) 2 _ _ = _ = ) 0 = = - = x -4 = 4-4 = 4 4 = 8. ( 2 v ) - = ( 2 (9) ) - = 6-6 = = 26 = = (0 - d) 0 = (0 - ) 0 = (-) 0 = 40. 0m - n -5 = 0(0) - ( -2) -5 = 0 _ 0 ( -2) 5 = 0 0 (-2)(-2)(-2)(-2)(-2) = Holt McDougal Algebra

3 4. (ab) -2 = ( ( -2 2 ) ) (8) = 2-2 = 2 2 = 2 2 = w v x v = 4() 0 ( -5) 0 = 4 = 4 4. k -4 = 44. 2z -8 = 2 z -8 k _ = 2b - 2 b - = 2 b = b x - = -5 x - = -5 5 x = - x 49. _ 2f 0 = 2 7g -0 7 f 0 g -0 = 2 7 g 0 = 2g s 5 = s 5 t -2 t -2 = s 5 t 2 = s 5 t 2 5. b 0 c 0 = b 0 c 0 = = _ = q -2 r q -2 r 0 s a -7 b 2 c d -4 = q 2 = q 2 s 0 = a -7 b 2 = a 7 = b 2 d 4 a 7 c b 2 = 2 z 8 = 2 z c -2 d = c -2 d = d c d 2 = c x -6 y -2 = 4 x -6 y -2 = 4 x 6 y 2 = 4 x 6 y r -5 = r -5 s - s - = s r 5 _ = s r w -5 = w -5 x -6 x -6 = x 6 w c d -4 d 4 c = x 6 w 5 m - n 5 = 2 = 2 m - n 5 m n 5 = 2n 5 m 57. _ h k - = 6m 2 6 h k - m 2 = 6 h k m 2 = _ h 6m 2 k 58. z -5 = = = 2 = 60. ( yz) 0 = ((-)(2)) 0 = (-2) 0 = 62. ( xy - ) -2 = (()(-) - ) -2 = (-6) -2 = _ ( -6) 2 = (-6)(-6) = ( yz) -x = ((-)(2)) - = (-2) - = _ ( -2) = (-2)(-2)(-2) = ( x + y) -4 = ( + (-)) -4 = = = 6 = 6. ( xyz) - = (()(-)(2)) - = (-6) - = (-6) = x -y = -(-) = = 65. xy -4 = ()(-) -4 = _ ( -) 4 = (-)(-)(-)(-) = = 66. Equation A is incorrect because 5 was incorrectly moved to the denominator. The negative exponent applies only to the base x. 67. a b -2 = a b -2 = a b 2 = a b v 0 w 2 y - = v 0 w 2 y - = w 2 y = w 2 y 68. c -4 d = c -4 d = d c 4 = d c ( a 2 b -7 ) 0 = 97 Holt McDougal Algebra

4 7. -5y -6 = -5 y -6 = -5 y 6 5 = - y a = 2 a b - b - = 2 a b = 2a b 75. x -8 = y 2 x -8 y 2 = x 8 y 2 = x 8 y p - = q - 5 p - q - = -4 p q = - 4q p _ 72. 2a -5 = 2 a -5 b -6 b -6 = 2 b 6 a 5 = 2b 6 a m 2 = m 2 n - n - = m 2 n = m 2 n 77. Red blood cell: 25,000 - = 25,000 White blood cell: (500) -2 = = = = 250,000 = 250,000 Platelet: (000) -2 = = _ = = _,000,000 = _,000, always 79. never 80. sometimes 8. sometimes 82. never 8. sometimes = 2 2 = (2 2 2) = 8 8 = a n a -n = 2-2 = 2 2 = ( ) = 9 9 = 85. Possible answer: Look at the pattern below. As the exponent goes down by, the value is half of what it was before. 2 = 8, 2 2 = 4, 2 = 2, 2 0 =, 2 - = 2, 2-2 = 4, 2 - = 8 = = 2 2 = = 2-2 ; = = 9 9 = ; = 8 8 = = 8-2 ; = = ; = = = ; = = = 0 - ; = 4 2 = 4 4 = 6 = ; = ; - 94a. fw = v b. fw = v fw = v f f test prep w = v f w = v f w = v f - w = vf - c. s = s D; Since 0.04 = 25 = 5 5 = = 5-2, A, B, and 5 2 C are all equal and do not equal J 6-2 = 6 2 = A 98., or.25 _ a b -2 = a b -2 4 c - c - = a c b 2 = a c b (6 + 2) 0 = = = = =, or a n ; a -n = a n and b 0 = if b 0. So you have a n, or simply a n. 98 Holt McDougal Algebra

5 challenge and extend 00. x y = 2 x y Possible answer: y increases more rapidly as x increases. 0. n n ( -) n n = ; (-) n = - if n is odd, and (-) n = if n is even. 6-2 rational exponents check it out! a. 8 4 = 4 ÇÇ 8 = b = ÇÇ ÇÇ 256 = + 4 = 5 2a. 6 4 = 6 = ( 6 4 4) = ( 4 ÇÇ 6 ) = 2 = 8 c = 27 = ( 27 4 ) 4 = ( ÇÇ 27 ) 4 = 4 = 8 x b. 2 5 = = ( 5 2 5) 2 = ( 5 Ç ) 2 =. C = 72m 4 = 72(8) 4 = 72 ( 4 ÇÇ 8 ) 4 = 72 ( Ç 4 ) = 72 () = = 944 The panda needs 944 Calories per day. 4a. 4 ÇÇÇ x 4 y 2 = (x 4 y 2 ) 4 = ( x 4 ) 4 (y 2 ) 4 = ( x 4 4 ) ( y 2 4 ) = ( x ) (y ) = x y think and discuss ( b. xy 2) 2 5 x Ç 5 ( = xy 2) 2 x = ( x 2 ) ( y 2 2 ) ( x - ) = ( x 2 ) y ( x - ) = ( x 2 ) ( x - ) y = x 2 + (-) y = xy. Rewrite the expression as 25 to the power, all 0 raised to the power 5. Then simplify the exponent. Finally take the square root. 2. to _ 2 Fractional Exponent _ b n _ m b n exercises guided practice. 5 Definition _ n A number raised to the power of is equal to the nth root of that number. A number raised to the power of _ m n is equal to the nth root of that number raised to the mth power. _ 2 6 Numerical Example = 6 = 6 _ 2 6 = 6 = 6 = 26 ( ) 2. 8 = Ç 8 = = ÇÇ 6 = = 6 Ç 0 = = ÇÇ 27 = = ÇÇ 8 = = ÇÇ 26 = = 9 Ç = = 4 ÇÇ 625 = = ÇÇ 6 + Ç = 6 + = = 4 ÇÇ 8 + Ç 8 = + 2 = = ( 8 4) = ( 4 ÇÇ 8 ) = = = ( 25 ) 2 = ( ÇÇ 25 ) 2 = 5 2 = = Ç 8 + ÇÇ 64 = = = ÇÇ 25-4 Ç = 5 - = = ( 8 ) 5 = ( Ç 8 ) 5 = 2 5 = = ( 25 2) = ( ÇÇ 25 ) = 5 = Holt McDougal Algebra

6 = ( 6 2) = ( ÇÇ 6 ) = 6 = = 4 Ç = 4 Ç = 22. P = 4a 2 = 4(64) 2 = 4( ÇÇ 64 ) = 4(8) = 2 The perimeter is 2 m. 2. ÇÇ x 4 y 2 = (x 4 y 2 ) 2 = ( x 4 2 ) ( y 2 2 ) = x 2 y = x 2 y 25. ÇÇ x 6 y 6 = (x 6 y 6 ) 2 = ( x 6 2 ) ( y 6 2 ) = x y = x y 27. ( a 2) 2 Ç a 2 = ( a 2 2 ) ( a 2 ) 2 = ( a ) ( a 2 2 ) = a a = a + = a 2 ( 29. z ) _ Ç z 2 = z _ ( z 2 ) 2 _ z 2 2 = z = z = z = ( 64 ) 4 = ( ÇÇ 64 ) 4 = 4 4 = = Ç 0 2 = 4 Ç 0 = Ç z 4 = ( z 4 ) 4 = z 4 4 = z = z 26. ÇÇÇ a 2 b 6 = ( a 2 b 6 ) = ( a 2 ) ( b 6 ) 28. ( x practice and problem solving = a 4 b 2 = a 4 b 2 6 ) 4 Ç y 4 = ( x ) 6 (y 4 ) 4 = ( x 2 4 ) ( y 4) = x 2 y = x 2 y 0. ÇÇ x 6 y 9 x 2 = (x 6 y 9 ) x 2 ( = 6 x _ ) 9 ( y ) x 2 = x 2 y = y x = ÇÇ 00 = = 5 Ç =. 52 = ÇÇ 52 = = ÇÇ 729 = = 5 ÇÇ 2 = = ÇÇ 96 = = 8 ÇÇ 256 = _ 2 = ÇÇ 400 = = ÇÇ 25 + ÇÇ 8 = 25 ÇÇ ÇÇ = = 4 = 5 - = = ÇÇ 2-5 ÇÇ 24 = - = = ( Ç 4 ) = 2 = = ( 4 ÇÇ 256 ) = 4 = = ( ÇÇ 00 ) = 0 = = ( Ç 9 ) 5 5. B = 8 = 8 = 5 = 24 2 m 2 (64) = 4 ÇÇ Ç 0 = = = ( ÇÇ 27 ) 2 = = = ( 6 ÇÇ 64 ) 5 = 2 5 = = ( Ç ) 5 = 5 = = ( 5 ÇÇ 24 ) 2 = 8 ( ÇÇ 64 ) 2 = 8 (4) 2 = 8 (6) = 2 The mass of the mouse s brain is 2g. 52. ÇÇ a 6 c 9 = ( a 6 c 9 ) = ( a 6 ) ( c 9 ) = a 2 c = a 2 c ÇÇÇ x 6 y 4 = (x 6 y 4 ) 4 = ( x 6 4 ) ( y 4 4 ) 56. ( x = x 4 y = x 4 y 2 = 2 = 9 5. ÇÇ 8m = ( 8m ) = ( 8 ) ( m ) = ( Ç 8 ) m = 2m 55. ÇÇ 27x 6 = ( 27x 6 ) = ( 27 ) ( x 6 ) = ( ÇÇ 27 ) x 2 = x 2 2y ) Ç x ( a 2 b 4 ) 2 Ç b 6 = ( x 2 ) 2 (y 2 ) x 2 = ( a 2 4 ) ( b 2) ( b 6 ) = x y 6 x = ( a ) ( b 2 6 ) ( b ) = x + y 6 = x 2 y 6 = x 2 y 6 = a b 2 b 2 = a b = a b 4 = ab Holt McDougal Algebra

7 58. x ÇÇ 6 y 6 yx 2 = (x 6 y 6 ) yx 2 = ( x 6 ) ( y 6 ) y - x -2 = ( x 2 ) (y 2 ) (y - ) ( x -2 ) = x 2-2 y 2 - = x 0 y = y ( 59. a 2 b 2) 4 Ç b 2 = ( a 2 4 ) ( b b 2 4 ) = ( a 8 ) ( b 2 ) ( b - ) = a 8 b 2 - = a 8 b = a 8 b x 4 = 4 ( 4 ÇÇ 256 ) x = 4 4 x = 4 x = x= 5 ( 225 x) x = 5 x 225 = 5 x 5 2 = 5 x x = x = 6 ( ÇÇ 64 ) x = 6 4 x = 6 x = x = 8 ( 27 4 x ) x 4 = 8 x ( 8 27 = ( 4 ÇÇ 8 ) x 27 = x x = 69 ) 2 = ÇÇ 8 69 ÇÇ = _ 8 ÇÇ 69 = 9 8 ) 4 = 4 ÇÇ ( 256 _ = 4 ÇÇ ÇÇ = 4 6. x 5 = ( x 5) 5 = 5 x = 6. x 6 = 0 ( x 6) 6 = 0 6 x = x 4 = 25 ( x 4 ) 4 = 25 4 x = ( ÇÇ 25 ) 4 x = 5 4 x = x 2 = 26 ( ÇÇ 6 ) x = 26 6 x = 26 x = 27) = ÇÇ 8 27 = Ç 8 ÇÇ 27 = 2 6) 2 = ÇÇ ( 8 7. ( = Ç ÇÇ 6 = 4 6) 72. ( ( ( ( 27 2 = ( ÇÇ 9 6 ) = ( Ç 9 ÇÇ 6 ) = ( 4) = ) 4 = ( 4 ÇÇ 6 8 ) = ( 4 ÇÇ 6 4 ÇÇ 8 ) = ( 2 ) = ) 2 = ( ÇÇ 4 25 ) = ( Ç 4 ÇÇ 25 ) 5 ) = ) = ÇÇ = ( 2 2 = ( ÇÇ ÇÇ ) = ( 4 ) 2 = ) 7. ( ( 4 49 ) 77. ( = ( ÇÇ ) = ( Ç ÇÇ ) = ( 2 ) 2 = 4 9 _ 2 = ( ÇÇ 4 49 ) = ( _ Ç 4 ÇÇ 49 ) = ( 2 7 ) = ) 4 = ( 4 ÇÇ 8 ) 4 25 ) 79. ( 8 = ( 4 Ç = ( 4 8 ÇÇ ) ) = 27 = ( ÇÇ ) _ = ( Ç 8 = ( 2 4 ÇÇ 25 ) 5) 4 = Lion: Wolf: L = 2m 5 L = 2m 5 = 2(24) 5 = 2(2) 5 = 2( 5 ÇÇ 24 ) = 2( 5 ÇÇ 2 ) = 2() = 6 = 2(2) = 24 The lion s lifespan is 6-24 = 2 years longer than the wolf s. 8. r = 0.62V = 0.62(27) = 0.62( ÇÇ 27 ) = 0.62() =.86 The radius is.86 in. 82. ( _ b ) = b _ = b = b. Also, by definition ( Ç b ) = b. Therefore b _ = b Ç. 8. n 2 will be less than n because 2 <. n 2 will be greater than n because 2 >. 84. A is incorrect; the first line should be 64 _ 2 = ( ÇÇ 64 ). 20 Holt McDougal Algebra

8 B) 85a. d = ( 0.8 L 2 _ = ( 0.8 ( ) ) = (0.8(25)) 2 = (00) 2 2 = ÇÇ 00 = 0 Distance to light source is 0 in. B) 2 b. d = ( 0.8 L = ( 0.8 ( ) ) = (0.8(500)) 2 = (400) 2 2 = ÇÇ 400 = 20 Distance doubles to 20 in = 4 2 = ( 4 ) 2 = 64 2 = = 4 2 = ( 4 2) = 2 = 8 It is often easier to take the square root first so that the remaining numbers in the calculation are smaller. 87. B; = 9 Ç + 8 Ç = + 2 = C; ÇÇ a 9 b = ( a 9 b ) = ( a 9 ) ( b ) = a b = a b challenge and extend 9. ( a ) ( a ) ( a ) = a ( + + ) = a = a 92. ( x 2) 5 ( x 2) = ( 5 x 2) ( x 2) = x ( ) = x 8 2 = x F; 4 2 = ( Ç 4 ) = 2 = H; ÇÇ 6 2 = ( Ç 2 4 ) 2 = ( ) = = 2 8 which is not an integer 9. ( x ) 4 ( x 5 ) = ( 4 x ) ( 5 x ) = x ( ) = x 9 = x 94. y 5 = 2 (y 5 ) 5 = 2 5 y 5 5 = 5 ÇÇ 2 y = 2 y = = 8 x (8) = (8) 8 x 8 = x 8 = ( x ) Ç 8 = x 2 = x 2 = x 97. S = (4π) (V) 2 = (4π) ((6π)) 2 = (4π) (08π) 2 = 4 π 08 2 π 2 = π + 2 = ( 2 2 ) 08 2 π = π = (2 08) 2 π = 26 2 π = ( ÇÇ 26 ) 2 π = 6 2 π = 6π cm x = x 27 = x = 27 ( x ) = 27 x = ÇÇ 27 x = x = Both volume and surface area are described by 6π (although the units are different). ready to go on? Section A Quiz. t -6 = 2-6 = 2 6 = = n - = (-5) - = _ ( -5) = (-5)(-5)(-5) = _ -25 = Holt McDougal Algebra

9 . r 0 s -2 = = 0 2 = 0 0 = x 4 = x 4 y -6 y -6 = x 4 y 6 = x 4 y 6 7. a - = a - b -2 b -2 = b 2 a = b 2 a 4. 5k - = 5 k - = 5 5 k = k 6. 8f -4 g 0 = 8 f -4 g 0 _ = 8 8 f 4 = f = = = 000 = = = = = = 0 0 = 0. 0 = = 0 0 = 00 0 = = = ÇÇ 8 = = ÇÇ 25 = = Ç 4 = 64 ÇÇ = = 0. ÇÇ x 8 y 4 = (x 8 y 4 ) 2 = ( x 8 ) 2 (y 4 ) 2 = ( x 8 2 ) ( y 4 2 ) = ( x 4 ) (y 2 ) = x 4 y 2 4. Ç r 9 = ( r 9 ) = r 9 = r 5. 6 z ÇÇ 2 = ( z 2 ) 6 = z 2 6 = z 2 6. ÇÇÇ p q 2 = (p q 2 ) = (p ) (q 2 ) = ( p ) ( q 2 ) = (p )(q 4 ) = pq 4 6- polynomials Check it out! a. The degree is. b. The degree is. c. The degree is. 2a. 5x: degree -6: degree 0 The degree of the polynomial is. b. x y 2 : degree 5 x 2 y : degree 5 -x 4 : degree 4 2: degree 0 The degree of the polynomial is 5. a. 6-4x 2 + x 5 + 9x x 5 + 9x - 4x The leading coefficient is. b. 8y 5 - y 8 + 4y -y 8 + 8y 5 + 4y The leading coefficient is -. 4a. Degree: Terms: 4 x + x 2 -x + 2 is a cubic polynomial. b. Degree: 0 Terms: 6 is a constant monomial. c. Degree: 8 Terms: -y 8 + 8y 5 + 4y is an 8th-degree trinomial t t + 6 = -6(5) (5) + 6 = -6(25) + 400(5) + 6 = = 606 When the firework explodes, it will be 606 ft above the ground. think and discuss. Possible answer: 2x 2 + x - contains an a expression with a negative exponent. - b contains a variable within a denominator. 2. Monomials x 2 exercises Polynomials Trinomials 2x 2 + 6x - 7 guided practice Binomials x + 2. d 2. c. a 4. The degree is The degree is. 6. The degree is The degree is Holt McDougal Algebra

10 8. x 2 : degree 2-2x: degree : degree 0 The degree of the polynomial is a 2 b: degree 2a b 5 : degree 8 The degree of the polynomial is y: degree -84y : degree 00: degree 0 -y 2 : degree 2 The degree of the polynomial is.. r : degree r 2 : degree 2-5: degree 0 The degree of the polynomial is. 2. a : degree a 2 : degree 2-2a: degree The degree of the polynomial is.. k 4 : degree 4 k : degree -2k 2 : degree 2 k: degree The degree of the polynomial is b b 2 b 2-2b + 5 The leading coefficient is. 5. 9a 8-8a 9-8a 9 + 9a 8 The leading coefficient is s 2 - s + - s 7 -s 7 + 5s 2 - s + The leading coefficient is x + x 2 - x 2 + 2x - The leading coefficient is. 8. 5g g 2 g 2 + 5g - 7 The leading coefficient is. 9. c 2 + 5c 4 + 5c - 4 5c 4 + 5c + c 2-4 The leading coefficient is Degree: 2 Terms: x 2 + 2x + is a quadratic trinomial. 2. Degree: Terms: 2 x - 7 is a linear binomial. 22. Degree: 4 Terms: 8 + k + 5k 4 is a quartic trinomial. 2. Degree: 4 Terms: 4 q q + q 4 is a quartic polynomial. 24. Degree: Terms: 2 5k 2 + 7k is a cubic binomial. 25. Degree: 4 Terms: 2a + 4a 2 - a 4 is a quartic trinomial r 2 +.4rl =.4(6) 2 +.4(6)(0) =.4(6) +.4(6)(0) = = 0.44 The surface area of the cone is approximately 0.44 cm 2. practice and problem solving 27. The degree is The degree is. 29. The degree is The degree is 0.. The degree is The degree is 5.. The degree is. 4. The degree is a 2 : degree 2 a 4 : degree 4-6a: degree The degree of the polynomial is b: degree -5: degree 0 The degree of the polynomial is. 7..5y 2 : degree 2-4.y: degree -6: degree 0 The degree of the polynomial is f 4 : degree 4 2f 6 : degree 6 0f 8 : degree 8 The degree of the polynomial is n : degree -2n: degree The degree of the polynomial is r : degree 4r 6 : degree 6 The degree of the polynomial is t - 4t 2 + t 4.9t - 4t 2 + t The leading coefficient is a - 0a a 2 + 8a + 2 The leading coefficient is x 7 - x + x - x 5 + x 0 x 0 + x 7 - x 5 + x - x The leading coefficient is m m 2 -m 2 - m + 7 The leading coefficient is x 2 + 5x x 5x + x 2 + 5x - 4 The leading coefficient is n + - n 2 -n 2-2n + The leading coefficient is d + d 2 - d + 5 -d + d 2 + 4d + 5 The leading coefficient is s 2 + 2s + 6 2s + s The leading coefficient is x 2 - x 5 - x + -x 5 - x + 4x 2 + The leading coefficient is Degree: 0 Terms: 2 is a constant monomial. 5. Degree: Terms: 6k is a linear monomial. 52. Degree: Terms:.5x - 4.x - 6 is a cubic trinomial. 5. Degree: 2 Terms: 4g + 2g 2 - is a quadratic trinomial. 54. Degree: 2 Terms: 2 2x 2-6x is a quadratic binomial. 55. Degree: 4 Terms: 6 - s - s 4 is a quartic trinomial. 204 Holt McDougal Algebra

11 56. Degree: Terms: c c is a cubic trinomial. 57. Degree: 2 Terms: -y 2 is a quadratic monomial v v 2 =.675(0) (0) 2 =.675(0) (900) = = The stopping distance of a car traveling at 0 mi/h is ft. 59. always 60. sometimes 6. never 62. sometimes 6a. 4c - 9c c = 4() - 9() () = 4() - 9() + 9.5() = = 58.5 The volume of the box when c = in. is 58.5 in. b. 4c - 9c c = 4(.5) - 9(.5) (.5) = 4(.75) - 9(2.25) + 9.5(.5) = = 66 The volume of the box when c =.5 in. is 66 in. c. 4c - 9c c = 4(4.25) - 9(4.25) (4.25) = 4(76.765) - 9(8.06) + 9.5(4.25) = = 0 The volume of the box when c = 4.25 in. is 0 in. d. Yes; the width of the cardboard is 8.5 in., so 4.25 in. cuts will meet, leaving nothing to fold up. Polynomial x = -2 x = 0 x = x x 5 + x + 4x x Possible answer: x 2 + x Possible answer: 5x Possible answer: Possible answer: 6x 7. Possible answer: x Possible answer: 2x 2 - x Possible answer: First identify the degree of each term. From left to right, the degrees are, 0, 2, 4, and. Arrange the terms in order of decreasing degree, and move the plus or minus sign in front of each term with it: -2x 4 + 4x + 5x 2 - x -. 74a. 2x: degree 6: degree 0 The degree of the polynomial is. 74b. 8x 2 : degree 2 2x: degree The degree of the polynomial is A is incorrect. The student incorrectly multiplied - by -2 before evaluating the power. test prep 76. C; A has degree 8, B has degree, C has degree 0, and D has degree 2. So C has the greatest degree. 77. J -x + 4x 2-5x + 7 = -(-) + 4(-) 2-5(-) + 7 = -(-) + 4() - 5(-) + 7 = = Time (s) Height (ft) The rocket will be the highest after 2 s. Challenge and Extend 79a. 0.06m m m = 0.06(2) (2) (2) = 0.06(8) (4) (2) = m m m = 0.06(5) (5) (5) = 0.06(25) (25) (5) = The average length of a two-month-old baby boy is 58 cm and the average length of a five-month-old baby boy is 65 cm. b. 0.06m m m = 0.06(0) (0) (0) = 0.06(0) (0) (0) = = 50.0 The average length of a newborn baby boy is 50.0 cm. c. The first three terms of the polynomial will equal 0, so just look at the constant. 80a. 4x 5 + x 6-4 b. yes; 0 < x < ; raising a number between 0 and to a higher power results in a lesser number. So if x is between 0 and, the bionomial with the least degree will have the greatest value. adding and subtracting polynomials Check it out! a. 2s 2 + s 2 + s = 5s 2 + s b. 4z z = 4z 4 + 6z = 20z Holt McDougal Algebra

12 c. 2x 8 + 7y 8 - x 8 - y 8 = 2x 8 - x 8 + 7y 8 - y 8 = x 8 + 6y 8 d. 9b c 2 + 5b c 2 - b c 2 = b c 2 2. (5a + a 2-6a + 2a 2 ) + ( 7a - 0a ) = (5a + 7a ) + (a 2 + 2a 2 ) + (-6a - 0a) = 2a + 5a 2-6a. (2x 2 - x 2 + ) - (x 2 + x + ) = (2x 2 - x 2 + ) + ( -x 2 - x - ) = ( 2x 2 - x 2 - x 2 ) + (-x) + ( - ) = -2x 2 - x 4. (-0.0x x - 500) + (-0.02x 2 + 2x - 700) -0.05x x think and discuss. -2x 2 and -9x 2 ; -4.7y and y; 5 x 2 y and 5x 2 y 2. Take the opposite of each term: -9t 2 + 5t Adding: (8 a 2 b + 9 a 2 + b ) + (7 a 2 b + 6 a b ) = 25 a 2 b + 5 a 2 + b exercises guided practice. 7a 2-0a 2 + 9a = -a 2 + 9a Polynomials Subtracting: (6 m 5 n - 8 m + 2) - (2 m 5 n + m - ) = (6 m 5 n - 8 m + 2) + (- 2 m 5 n - m + ) = 4 m 5 n - 9 m + 2. x 2 + 9y 2-6x 2 = x 2-6x 2 + 9y 2 = 7x 2 + 9y r r + 0.9r 4 = 0.07r r r = 0.26r r 4. 4 p + 2 p 5. 5b c + b c - b c = 2 p = b c 6. -8m m = -8m + m = m - 7. (5n + n + 6) + (8n + 9) = (5n + 8n ) + n + (6 + 9) = 2n + n (.7q 2-8q +.7) + (4.q 2-2.9q +.6) = (.7q q 2 ) + (-8q - 2.9q) + (.7 +.6) = 8q 2-0.9q (-x + 2) + (9x 2 + 2x - 8) = 9x 2 + (-x + 2x) + (2-8) = 9x 2 - x (9x 4 + x ) + (2x 4 + 6x - 8x 4 + x ) = (9x 4 + 2x 4-8x 4 ) + (x + 6x + x ) = x 4 + 8x. (6c 4 + 8c + 6) - ( 2c 4 ) = (6c 4 + 8c + 6) + ( -2c 4 ) = ( 6c 4-2c 4 ) + 8c + 6 = 4c 4 + 8c (6y 2-8y + 9) - (6y 2-2y + 7y) = (6y 2-8y + 9) + (-6y 2 + 2y - 7y) = (6y 2-6y 2 ) + (-8y + 2y - 7y) + 9 = 0y 2 - y + 9. (2r + 5) - (5r - 6) = (2r + 5) + (-5r + 6) = (2r - 5r) + (5 + 6) = -r + 4. (-7k 2 + ) - (2k 2 + 5k - ) = (-7k 2 + ) + (-2k 2-5k + ) = ( -7k 2-2k 2 ) + (-5k) + ( + ) = -9k 2-5k m ABD = (8a 2-2a + 5) + (7a + 4) = 8a 2 + (-2a + 7a) + (5 + 4) = 8a 2 + 5a + 9 practice and problem solving 6. 4k + 6k 2 + 9k = 4k + 9k + 6k 2 = k + 6k m + 2n 2 + 6n - 8m = 5m - 8m + 2n 2 + 6n = 2n 2 + 6n - m a 4-8.b 4 -.6b 4 = 2.5a 4 -.7b xy - 4x 2 y - 2xy = 7xy - 2xy - 4x 2 y = -4x 2 y + 5xy 2. -6x + 5x + 2x + 4x = -6x + 2x + 4x + 5x = 5x 22. x 2 + x + x + 2x 2 = x 2 + 2x 2 + x + x = x 2 + 4x 9. 2d d 5 = 2d 5 - d 5 + = d x x - = x - x = 2x Holt McDougal Algebra

13 24. b - 2b - - b - b = b - b - 2b - b - = 2b - b ( 2t 2-8t ) + (8t 2 + 9t) = (2t 2 + 8t 2 ) + (-8t + 9t) = 0t 2 + t 26. (-7x 2-2x + ) + ( 4x 2-9x ) = (-7x 2 + 4x 2 ) + (-2x - 9x) + = -x 2 - x ( x 5 - x ) + (x 4 + x) = (x 5 + x 4 ) + (-x + x) = x 5 + x (-2z + z + 2z + z)+ ( z - 5z 2 ) = (-2z + 2z + z ) + ( -5z 2 ) + (z + z) = z - 5z 2 + 2z 29. (t + 8t 2 ) - ( t ) = (t + 8t 2 ) + ( -t ) = ( t - t ) + 8t 2 = -2t + 8t 2 0. ( x 2 - x ) - (x 2 + x - x) = ( x 2 - x ) + (-x 2 - x + x) = ( x 2 - x 2 ) + (-x - x + x) = 2x 2 - x. (5m + ) - ( 6m - 2m 2 ) = (5m + ) + (-6m + 2m 2 ) = -6m + 2m 2 + 5m + 2. (s 2 + 4s)- (-0s 2 + 6s) = (s 2 + 4s)+ ( 0s 2-6s ) = (s 2 + 0s 2 ) + (4s - 6s) = s 2-2s. width = (6w 2 + 8) - 2(w 2 - w + 2) = (6w 2 + 8) + (-2( w 2 ) - 2(-w) - 2(2)) = (6w 2 + 8) + (-2w 2 + 6w - 4) = ( 6w 2-2w 2 ) + 6w + (8-4) = 4w 2 + 6w P = 2l + 2w = 2(4a + b) + 2(7a - 2b) = 2(4a) + 2(b) + 2(7a) + 2(-2b) = 8a + 6b + 4a - 4b = 8a + 4a + 6b - 4b = 22a + 2b 5. (2t - 7) + (-t + 2) = (2t - t) + (-7 + 2) = t (4m 2 + m)+ ( -2m 2 ) = ( 4m 2-2m 2 ) + m = 2m 2 + m 7. (4n - 2) - 2n = (4n - 2) + (-2n) = (4n - 2n) + (-2) = 2n (4x 2 + x - 6) + (2x 2-4x + 5) 8. (-v - 7) - (-2v) = (-v - 7) + (2v) = (-v + 2v) + (-7) = v - 7 = (4x 2 + 2x 2 ) + (x - 4x) + (-6 + 5) = 6x 2 - x ( 2z 2 - z - ) + ( 2z 2-7z - ) = (2z 2 + 2z 2 ) + (-z - 7z) + (- - ) = 4z 2-0z (5u 2 + u + 7) - (u + 2u 2 + ) = (5u 2 + u + 7) + ( -u - 2u 2 - ) = ( -u ) + ( 5u 2-2u 2 ) + u + (7 - ) = -u + u 2 + u (-7h 2-4h + 7) - (7h 2-4h + ) = (-7h 2-4h + 7) + (-7h 2 + 4h - ) = ( -7h 2-7h 2 ) + (-4h + 4h) + (7 - ) = -4h P = 2l + 2w 5 = 2(2x + ) + 2(x + 7) 5 = 2(2x) + 2() + 2(x) + 2(7) 5 = 4x x = 4x + 6x = 0x = 0x 5 0 = 0x 0 = x, or x = Yes; the simplified form of both expressions is 5m 2 + 2m - 0. No; the simplified form of the original expression is -9m 2-2m + 0 and the simplified form of the new expression is -9m 2 + 2m B is incorrect. The student incorrectly tried to combine 6n and -n 2, which are not like terms, and tried to combine 4n 2 and 9n, which are not like terms. Polynomial Polynomial 2 Sum 46. x 2-6 x 2-0x + 2 4x 2-0x x + 5 x + 6 5x x 4 - x 2-9 5x x 4 - x x - 6x - 6x + 4 7x x + 5x 2 7x - 5x 2 + 9x x 2 + x - 5 x + x x 2 + 2x No; polynomial addition simply involves combining like terms. No matter what order the terms are combined in, the sum will be the same. Yes; in polynomial subtraction, the subtraction sign is distributed among all terms in the second polynomial, changing all the signs to their opposites. 207 Holt McDougal Algebra

14 5a. x + 4 x Possible answer: 2m + m, m + m + m + m 6. Possible answer: 4m + m b. P = 2l + 2w = 2(x + 4) + 2(x - ) = 2(x) + 2(4) + 2(x) + 2(-) = 2x x - 6 = 2x + 2x = 4x + 2 c. P = 4x + 2 = 4(5) + 2 = = 62 He will need 62 ft of fencing. test prep 54. C; Since -4y 2 + 9y 2 + 2y 2 = -y 2, and - 2 =, the term must be in the form ay. So -2y + ay - 6y = -5y gives -2 + a - 6 = -5 or a =. So the missing term is y. 55. G; Since 2t - 4t - (-7t - t) = 2t + 6t -5t - t, G is correct. 56a. P = 2l + 2w - = 2(2x - ) + 2(x + 4) - = 2(2x) + 2(-) + 2(x) + 2(4) - = 4x x = 4x + 2x = 6x + b. 6x + = 50 _ - - 6x = 47 6x 6 = 47 6 x 7.8 7; If x = 7, Tammy will need 6(7) + = 45 feet of wallpaper border. However, if x = 8, Tammy will need 6(8) + = 5 feet of wallpaper border, which is more than the store has. c. (2x - ) ft (x + 4) ft = (2(7) - )ft (7 + 4) ft = ft ft CHALLENGE AND EXTEND 57. P = b + 2s - 2s - 2s P - 2s = b b = (2x + x 2 + 8) - 2(x + 5) = (2x + x 2 + 8) + (-2x - 2(5)) = (2x + x 2 + 8) + ( -2x - 0 ) = ( 2x - 2x ) + x 2 + (8-0) = x Possible answer: 2m + 2m, 2m + m 59. Possible answer: 5m + 2m, m - m 62. Possible answer: 2m + m 2 + m, m + m 2 + m, m - 2m 2 + m 6-5 multiplying POlynomials Check it out! a. ( x ) ( 6x 2 ) = ( 6)( x x 2 ) = 8x 5 b. ( 2r 2 t ) ( 5t ) c. ( x 2 y ) ( 2x z 2 ) (y 4 z 5 ) = ( 2 ) ( x 2 x ) (y y 4 )( z 2 z 5 ) = 4x 5 y 5 z 7 2a. 2(4x 2 + x + ) = 2( 4x 2 ) + 2(x) + 2() = 8x 2 + 2x + 6 b. ab(5a 2 + b) = ab( 5a 2 ) + ab(b) = ( 5)( a a 2 ) ( b) + ()(a)(b b) = 5a b + ab 2 c. 5r 2 s 2 ( r - s) = 5r 2 s 2 ( r) + 5r 2 s 2 ( -s) = (2 5)( r 2 ) ( t t ) = 0r 2 t 4 = (5)( r 2 r ) ( s 2 ) + (5 (-)) ( r 2 ) ( s 2 s ) = 5r s 2-5r 2 s a. (a + )(a - 4) = a(a) + a(-4) + (a) + (-4) = a 2-4a + a - 2 = a 2 - a - 2 b. ( x - ) 2 = (x - )(x - ) = x(x) + x(-) - (x) - (-) = x 2 - x - x + 9 = x 2-6x + 9 c. ( 2a - b 2 ) (a + 4b 2 ) = 2a(a) + 2a( 4b 2 ) - b 2 ( a) - b 2 ( 4b 2 ) = 2a 2 + 8ab 2 - ab 2-4b 4 = 2a 2 + 7ab 2-4b 4 4a. (x + )(x 2-4x + 6) = x(x 2-4x + 6) + (x 2-4x + 6) = x( x 2 ) + x(-4x) + x(6) + ( x 2 ) + (-4x) + (6) = x - 4x 2 + 6x + x 2-2x + 8 = x - x 2-6x Holt McDougal Algebra

15 b. (x + 2)(x 2-2x + 5) = x(x 2-2x + 5) + 2(x 2-2x + 5) = x( x 2 ) + x(-2x) + x(5) + 2 ( x 2 ) + 2( -2x) + 2(5) = x - 6x 2 + 5x + 2x 2-4x + 0 = x - 4x 2 + x + 0 5a. Let x represent the width of the rectangle. A = lw = (x - 4)(x) = x(x) - 4(x) = x 2-4x The area is represented by x 2-4x. b. A = x 2-4x = (6) 2-4(6) = 6-24 = 2 The area is 2 m 2. think and discuss. Possible answer: Both numbers and polynomials are set up in two rows and require you to multiply each item in the top row by an item in the bottom row. In the end, you add vertically to get the answer. When you are multiplying polynomials, the items are monomial terms. When your are multiplying numbers, the items are digits. 2. Distributive Property: 5x (x + 2) = 5x 2 + 0x Rectangle model: (x + 2)(x 2 + 2x + ) x + 2 x 2 x + 2x + 2 x 2 4x 2 x 2 x 2 x + 4x 2 + 5x + 2 exercises guided practice. ( 2x 2 ) ( 7x 4 ) = (2 7)( x 2 x 4 ) = 4x 6 Multiplying Polynomials 2. ( -5mn ) ( 4m 2 n 2 ) FOIL method: (x + )(x + 2) = x 2 + 2x + x + 2 = x 2 + x + 2 Vertical method: (x + 2)(x 2 + x + 2) x 2 + x + 2 x x x x + x x x + 5 x x + 4 = (-5 4)( m m 2 ) ( n n 2 ) = -20m n 5. ( 6rs 2 ) ( s t 2 ) ( 2 r 4 t ) = ( 6 2 ) ( r r 4 ) ( s 2 s ) ( t 2 t ) = r 5 s 5 t 5 4. ( a 5 ) (2a) = ( 2 ) ( a 5 a ) = 4a 6 5. (-x 4 y 2 )(-7x y) = (- (-7))( x 4 x ) (y 2 y) = 2x 7 y 6. (-2pq )(5p 2 q 2 )(-q 4 ) = (-2 5 (-))(p p 2 )(q q 2 q 4 ) = 0p q (x 2 + 2x + ) = 4( x 2 ) + 4(2x) + 4() = 4x 2 + 8x ab(2a 2 + b ) = ab( 2a 2 ) + ab ( b ) = ( 2)( a a 2 ) ( b) + ( )(a) ( b b ) = 6a b + 9ab a b(a 2 b + ab 2 ) = 2a b( a 2 b ) + 2a b ( ab 2 ) = (2 )( a a 2 ) ( b b) + (2) ( a a ) ( b b 2 ) = 6a 5 b 2 + 2a 4 b 0. -x(x 2-4x + 6) = -x( x 2 ) - x(-4x) - x(6) = -x + 2x 2-8x. 5x 2 y(2xy - y) = 5x 2 y(2xy ) + 5x 2 y(-y) = (5 2)( x 2 x ) (y y ) + (5 (-))( x 2 ) ( y y) = 0x y 4-5x 2 y m 2 n mn 2 (4m - n) = (5)( m 2 m ) ( n n 2 ) (4m - n) = 5m n 5 (4m - n) = 5m n 5 (4m) + 5m n 5 ( -n) = (5 4)( m m ) ( n 5 ) + (5 (-)) ( m ) ( n 5 n ) = 20m 4 n 5-5m n 6. (x + )(x - 2) = x(x) + x(-2) + (x) + (-2) = x 2-2x + x - 2 = x 2 - x ( x + ) 2 = (x + )(x + ) = x(x) + x() + (x) + () = x 2 + x + x + = x 2 + 2x Holt McDougal Algebra

16 5. ( x - 2) 2 = (x - 2)(x - 2) = x(x) + x(-2) - 2(x) - 2(-2) = x 2-2x - 2x + 4 = x 2-4x (y - )(y - 5) = y(y) + y(-5) - (y) - (-5) = y 2-5y - y + 5 = y 2-8y ( 4a - 2b ) ( a - b 2 ) = 4a ( a) + 4a ( -b 2 ) - 2b(a) - 2b ( -b 2 ) = 4a 4-2ab - 2a b 2 + 6b 8. ( m 2-2mn ) (mn + n 2 ) = m 2 (mn) + m 2 ( n 2 ) - 2mn(mn) - 2mn ( n 2 ) = m n + m 2 n 2-6m 2 n 2-2mn = m n - 5m 2 n 2-2mn 9. (x + 5)(x 2-2x + ) = x(x 2-2x + ) + 5(x 2-2x + ) = x( x 2 ) + x(-2x) + x() + 5 ( x 2 ) + 5(-2x) + 5() = x - 2x 2 + x + 5x 2-0x + 5 = x + x 2-7x (x + 4)(x 2-5x + 2) = x(x 2-5x + 2) + 4(x 2-5x + 2) = x( x 2 ) + x(-5x) + x(2) + 4 ( x 2 ) + 4(-5x) + 4(2) = x - 5x 2 + 6x + 4x 2-20x + 8 = x - x 2-4x (2x - 4)(-x + 2x - 5) = 2x(-x + 2x - 5) - 4(-x + 2x - 5) = 2x( -x ) + 2x(2x) + 2x(-5) - 4 ( -x ) - 4(2x) - 4(-5) = -6x 4 + 4x 2-0x + 2x - 8x + 20 = -6x 4 + 2x + 4x 2-8x (-4x + 6)(2x - x 2 + ) = -4x(2x - x 2 + ) + 6(2x - x 2 + ) = -4x( 2x ) -4x ( -x 2 ) -4x() + 6 ( 2x ) + 6 ( -x 2 ) + 6() = -8x 4 + 4x - 4x + 2x - 6x = -8x 4 + 6x - 6x 2-4x (x - 5)(x 2 + x + ) = x(x 2 + x + ) - 5(x 2 + x + ) = x( x 2 ) + x(x) + x() -5 ( x 2 ) - 5(x) - 5() = x + x 2 + x - 5x 2-5x - 5 = x - 4x 2-4x (a + b)(a - b)(b - a) = (a(a) + a(-b) + b(a) + b(-b))( b- a) = (a 2 - ab + ab - b 2 )( b - a) = ( a 2 - b 2 ) ( b - a) = a 2 ( b) + a 2 ( -a) - b 2 ( b) - b 2 ( -a) = a 2 b - a - b + ab 2 = -a + a 2 b + ab 2 - b 25a. A = lw = (2x - )(x) = 2x(x) - (x) = 2x 2 - x The area is represented by 2x 2 - x. b. A = 2x 2 - x = 2(4) 2 - (4) = 2(6) - (4) = 2-2 = 20 The area is 20 in 2. practice and problem solving 26. ( x 2 ) ( 8x 5 ) = ( 8)( x 2 x 5 ) = 24x ( -2r s 4 ) ( 6r 2 s ) = (-2 6)( r r 2 ) ( s 4 s ) = -2r 5 s (5xy 2 ) ( x 2 z ) (y z 4 ) = ( 5 ) ( x x 2 ) (y 2 y )( z z 4 ) = 5x y 5 z ( -2a ) ( -5a) = (-2 (-5))( a a ) = 0a 4 0. (6x y 2 )(-2x 2 y) = (6 (-2))( x x 2 ) (y 2 y) = -2x 5 y. ( -a 2 b ) ( -2b ) ( -a b 2 ) = (- (-2) (-))( a 2 a ) ( b b b 2 ) = -6a 5 b 6 2. ( 7x 2 ) (xy 5 )(2x y 2 ) = (7 2)( x 2 x x ) (y 5 y 2 ) = 4x 6 y 7. ( -4a bc 2 ) ( a b 2 c) ( ab 4 c 5 ) = (-4 )( a a a ) ( b b 2 b 4 ) ( c 2 c c 5 ) = -2a 7 b 7 c 8 4. ( 2mn 2 ) ( 2m 2 n ) (mn) = (2 2)( m m 2 m ) ( n 2 n n ) = 24m 4 n 4 20 Holt McDougal Algebra

17 5. 9s(s + 6) = 9s(s) + 9s(6) = 9s s 7. x( 9x 2-4x ) = x( 9x 2 ) + x(-4x) = 27x - 2x s 2 t ( 2s - t 2 ) = 5s 2 t (2s) + 5s 2 t ( -t 2 ) 6. 9( 2x 2-5x ) = 9( 2x 2 ) + 9(-5x) = 8x 2-45x 8. (2x 2 + 5x + 4) = ( 2x 2 ) + (5x) + (4) = 6x 2 + 5x + 2 = (5 2)( s 2 s ) ( t ) + (5 (-)) ( s 2 ) ( t t 2 ) = 0s t - 5s 2 t x 2 y 5x 2 y(6x + y 2 ) = (5)( x 2 x 2 ) (y y)(6x + y 2 ) = 5x 4 y 4 (6x + y 2 ) = 5x 4 y 4 (6x) + 5x 4 y 4 (y 2 ) = (5 6)( x 4 x ) (y 4 ) + (5)( x 4 ) (y 4 y 2 ) = 0x 5 y 4 + 5x 4 y x( 2x 2 - x - ) = -5x( 2x 2 ) - 5x(-x) - 5x(-) = -0x + 5x 2 + 5x a 2 b ( ab 2 - a 2 b ) = -2a 2 b ( ab 2 ) - 2a 2 b ( -a 2 b ) = (-2 )( a 2 a ) ( b b 2 ) - (2 -) ( a 2 a 2 ) ( b b ) = -6a b 5 + 2a 4 b x y x 2 y 2 (2x - y) = (-7)( x x 2 ) (y y 2 )(2x - y) = -7x 5 y (2x - y) = -7x 5 y (2x) - 7x 5 y ( -y) = (-7 2)( x 5 x ) (y ) + (-7 (-))( x 5 ) (y y) = -4x 6 y + 7x 5 y (x + 5)(x - ) = x(x) + x(-) + 5(x) + 5(-) = x 2 - x + 5x - 5 = x 2 + 2x ( x + 4) 2 = (x + 4)(x + 4) = x(x) + x(4) + 4(x) + 4(4) = x 2 + 4x + 4x + 6 = x 2 + 8x ( m - 5) 2 = (m - 5)(m - 5) = m(m) + m(-5) - 5(m) - 5(-5) = m 2-5m - 5m + 25 = m 2-0m (5x - 2)(x + ) = 5x(x) + 5x() - 2(x) - 2() = 5x 2 + 5x - 2x - 6 = 5x 2 + x (x - 4) 2 = (x - 4)(x - 4) = x(x) + x(-4) - 4(x) - 4(-4) = 9x 2-2x -2x + 6 = 9x 2-24x (5x + 2)(2x - ) = 5x(2x)+ 5x(-) + 2(2x)+ 2(-) = 0x 2-5x + 4x - 2 = 0x 2 - x (x - )(x - 2) = x(x) + x(-2) - (x) - (-2) = x 2-2x - x + 2 = x 2 - x (x - 8)(7x + 4) = x(7x) + x(4) - 8(7x) - 8(4) = 7x 2 + 4x - 56x - 2 = 7x 2-52x (2x + 7)(x + 7) = 2x(x) + 2x(7) + 7(x) + 7(7) = 6x 2 + 4x + 2x + 49 = 6x 2 + 5x (x + 2)(x 2 - x + 5) = x(x 2 - x + 5) + 2(x 2 - x + 5) = x( x 2 ) + x(-x) + x(5) + 2 ( x 2 ) + 2(-x) + 2(5) = x - x 2 + 5x + 2x 2-6x + 0 = x - x 2 - x (2x + 5)(x 2-4x + ) = 2x(x 2-4x + ) + 5(x 2-4x + ) = 2x( x 2 ) + 2x(-4x) + 2x() + 5 ( x 2 ) + 5(-4x) + 5() = 2x - 8x 2 + 6x + 5x 2-20x + 5 = 2x - x 2-4x (5x - )(-2x + 4x - ) = 5x(-2x + 4x - ) - (-2x + 4x - ) = 5x( -2x ) + 5x(4x) + 5x(-) - ( -2x ) - (4x) - (-) = -0x x 2-5x + 2x - 4x + = -0x 4 + 2x + 20x 2-9x (x - )(x 2-5x + 6) = x(x 2-5x + 6) - (x 2-5x + 6) = x( x 2 ) + x(-5x) + x(6) - ( x 2 ) - (-5x) - (6) = x - 5x 2 + 6x - x 2 + 5x - 8 = x - 8x 2 + 2x ( 2x 2 - ) (4x - x 2 + 7) = 2x 2 (4x - x 2 + 7) - (4x - x 2 + 7) = 2x 2 ( 4x ) + 2x 2 ( -x 2 ) + 2x 2 (7) - ( 4x ) - ( -x 2 ) - (7) = 8x 5-2x 4 + 4x 2-2x + x 2-2 = 8x 5-2x 4-2x + 7x Holt McDougal Algebra

18 58. ( x - 4) = (x - 4)(x - 4)(x - 4) = (x(x) + x(-4) - 4(x) - 4(-4))( x - 4) = (x 2-4x - 4x + 6)( x - 4) = (x 2-8x + 6)( x - 4) = (x - 4)(x 2-8x + 6) = x(x 2-8x + 6) - 4(x 2-8x + 6) = x( x 2 ) + x(-8x) + x(6) - 4 ( x 2 ) - 4(-8x) - 4(6) = x - 8x 2 + 6x - 4x 2 + 2x - 64 = x - 2x x (x - 2)(x 2 + 2x + ) = x(x 2 + 2x + ) - 2(x 2 + 2x + ) = x( x 2 ) + x(2x) + x() - 2 ( x 2 ) - 2(2x) - 2() = x + 2x 2 + x - 2x 2-4x - 2 = x - x (2x + 0)(4 - x + 6x ) = 2x(4 - x + 6x ) + 0(4 - x + 6x ) = 2x(4) + 2x(-x) + 2x( 6x ) + 0(4) + 0(-x) + 0( 6x ) = 8x - 2x 2 + 2x x + 60x = 2x x - 2x 2-2x ( - x) = ( - x)( - x)( - x) = (() + (-x) - x() - x(-x))( - x) = ( - x - x + x 2 )( - x) = ( - 2x + x 2 )( - x) = ( - x)( - 2x + x 2 ) = ( - 2x + x 2 ) - x( - 2x + x 2 ) = - 2x + x 2 -x() - x(-2x) -x( x 2 ) = - 2x + x 2 - x + 2x 2 - x = -x + x 2 - x + 62a. A = lw = (x + )(x) = x(x) + (x) = x 2 + x The area is represented by x 2 + x. b. A = x 2 + x = (5) 2 + (5) = = 40 The area is 40 ft A = s 2 = (4x - 6) 2 = (4x - 6)(4x - 6) = 4x(4x) + 4x(-6) - 6(4x) - 6(-6) = 6x 2-24x - 24x + 6 = 6x 2-48x + 6 The area is represented by 6x 2-48x a. x + 4 x + b. A = lw = (x + 4)(x + ) = x(x) + x() + 4(x) + 4() = x 2 + x + 4x + 4 = x 2 + 5x + 4 The area is represented by x 2 + 5x + 4. c. A = x 2 + 5x + 4 = (4) 2 + 5(4) + 4 = = 40 The area is 40 ft 2. A Degree of A B Degree of B A B Degree of A B 2x 2 2 x 5 5 6x a. 5x 2x x 5 + 5x 5 b. x x 2 - x 2 x 4 - x + 4 2x 2-2x c. x - x - 2x 2 + x 4-5x + 6x 2 + x - 4 d. m + n 66. A = lw = (2x + )(4x) = 2x(4x) + (4x) = 8x 2 + 2x The area is represented by 8x 2 + 2x. 67. A = lw = (2x + )(2x + ) = [(2x) + ()](2x + ) = (6x + )(2x + ) = 6x(2x) + 6x() + (2x) + () = 2x 2 + 6x + 6x + = 2x 2 + 2x + The area is represented by 2x 2 + 2x A = lw = (x - 5)(x - 5) = x(x) + x(-5) - 5(x) - 5(-5) = x 2-5x - 5x + 25 = x 2-0x + 25 The area is represented by x 2-0x a. A = lw = (2x)(x) = 2x 2 The area is represented by 2x 2. b. A = 2x 2 = 2(20) 2 = 2(400) = 800 The area is 800 m Holt McDougal Algebra

19 70. (.5a ) ( 4a 6 ) = (.5 4)( a a 6 ) = 6a 9 7. (2x + 5)(x - 6) = 2x(x) + 2x(-6) + 5(x) + 5(-6) = 2x 2-2x + 5x - 0 = 2x 2-7x (g - )(g + 5) = g(g) + g(5) - (g) - (5) = g 2 + 5g - g - 5 = g 2 + 4g (4x - 2y)(2x - y) = 4x(2x) + 4x(-y) - 2y(2x) - 2y(-y) = 8x 2-2xy - 4xy + 6y 2 = 8x 2-6xy + 6y (x + )(x - ) = x(x) + x(-) + (x) + (-) = x 2 - x + x - 9 = x (.5x - )(4x + 2) =.5x(4x) +.5x(2) - (4x) - (2) = 6x 2 + x - 2x - 6 = 6x 2-9x (x - 0)(x + 4) = x(x) + x(4) - 0(x) - 0(4) = x 2 + 4x - 0x - 40 = x 2-6x x 2 ( x + ) = x 2 ( x) + x 2 () = x + x (x + )(x 2 + 2x) = x( x 2 ) + x(2x) + ( x 2 ) + (2x) = x + 2x 2 + x 2 + 2x = x + x 2 + 2x 79. (x - 4)(2x 2 + x - 6) = x(2x 2 + x - 6) - 4(2x 2 + x - 6) = x( 2x 2 ) + x(x) + x(-6) - 4 ( 2x 2 ) - 4(x) - 4(-6) = 2x + x 2-6x - 8x 2-4x + 24 = 2x - 7x 2-0x (a + b)(a - b) 2 = (a + b)(a - b)(a - b) = (a(a) + a(-b) + b(a) + b(-b))( a - b) = (a 2 - ab + ab - b 2 )( a - b) = ( a 2 - b 2 ) ( a - b) = a 2 ( a) + a 2 ( -b) - b 2 ( a) - b 2 ( -b) = a - a 2 b - ab 2 + b 8. (2p - q) = (2p - q)(2p - q)(2p - q) = (2p(2p) + 2p(-q) - q(2p) - q(-q))( 2p - q) 82a. = (4p 2-6pq - 6pq + 9q 2 )( 2 p - q) = (4p 2-2pq + 9q 2 )(2p - q) = (2p - q)(4p 2-2pq + 9q 2 ) = 2p(4p 2-2pq + 9q 2 ) - q(4p 2-2pq + 9q 2 ) = 2p(4p 2 ) + 2p(-2pq) + 2p(9q 2 ) - q(4p 2 ) - q(-2pq) - q(9q 2 ) = 8p - 24p 2 q + 8pq 2-2p 2 q + 6pq 2-27q = 8p - 6p 2 q + 54pq 2-27q x x 25 b. The length is 25 + x + x = 2x The width is 0 + x + x = 2x + 0. c. A = lw = (2x + 25)(2x + 0) = 2x(2x) + 2x(0) + 25(2x) + 25(0) = 4x x + 50x = 4x x Possible answer: Each letter in FOIL represents a pair of terms in a certain position within the factors. The letters must account for every pairing of terms while describing first, outside, inside, and last positions. This is only possible with two binomials. 84. A = lwh = (x + 5)(x)(x + 2) = (x(x) + 5(x))( x + 2) = (x 2 + 5x)(x + 2) = x 2 ( x) + x 2 (2) + 5x(x) + 5x(2) = x + 2x 2 + 5x 2 + 0x = x + 7x 2 + 0x The area is represented by x + 7x 2 + 0x. 85. Yes; x = Let x represent the width of the rectangle. A = lw = (x + )(x) = x(x) + (x) = x 2 + x Since (4.5) = , the width of the rectangle is about 4.5 ft. test prep 87. C (a + )(a - 6) = a(a) + a(-6) + (a) + (-6) = a 2-6a + a - 6 = a 2-5a Holt McDougal Algebra

20 88. H 2a( a 2 - ) = 2a( a 2 ) + 2a(-) = 2a - 2a 89. D x y 2 z x 2 yz = ()( x x 2 ) (y 2 y)(z z) = x 5 y z 2 This has degree = 0. challenge and extend 90. 6x 2-2(x 2-2x + 4) = 6x 2-2( x 2 ) - 2(-2x) - 2(4) = 6x 2-6x 2 + 4x - 8 = 4x x 2-2x(x + ) = x 2-2x(x) - 2x() = x 2-2x 2-6x = -x 2-6x 92. x(4x - 2) + x(x + ) = x(4x) + x(-2) + x(x) + x() = 4x 2-2x + x 2 + x = 7x 2 + x 9a. A = lw = (x + )(x - ) = x(x) + x(-) + (x) + (-) = x 2 - x + x - = x 2 - The area is represented by x 2 -. b. A = lw = (x + 5)(x + ) - (x + )(x - ) = x(x) + x() + 5(x) + 5() - ( x 2 - ) = x 2 + x + 5x x 2 + = 8x A = s 2 = (8 + 2x) 2 = (8 + 2x)(8 + 2x) = 8(8) + 8(2x) + 2x(8) + 2x(2x) = x + 6x + 4x 2 = 4x 2 + 2x + 64 P = 4s = 4(x ) = 4( x 2 ) + 4(48) = 4x A = P 4x 2 + 2x + 64 = 4x x 2-4x 2 2x + 64 = x = 28 2x 2 = 28 2 x = x(x + )(x + 2) = (x(x) + x())( x + 2) = (x 2 + x)(x + 2) = x 2 ( x) + x 2 (2) + x(x) + x(2) = x + 2x 2 + x 2 + 2x = x + x 2 + 2x 96. x m (x n + x n - 2 ) = x 5 + x x m ( x n ) + x m ( x n - 2 ) = x 5 + x x m + n + x m + n - 2 = x 5 + x Therefore, it must be true that: m + n = 5 m + n = 5 m + n - 2 = m + n = 5 Therefore, the system is consistent and dependent, so there is an infinite number of solutions. One is m = 2; n = x a (5x 2a - + 2x 2a + 2 ) = 0x + 4x 8 2x a ( 5x 2a - ) + 2x a ( 2x 2a + 2 ) = 0x + 4x 8 0x a - + 4x a + 2 = 0x + 4x 8 Therefore, it must be true that: a - = and a + 2 = a = 6 and a = 6 a = 6 a = 6 a = special Products of Binomials Check it out! a. ( a + b) 2 = a 2 + 2ab + b 2 ( x + 6) 2 = (x) 2 + 2(x)(6) + (6) 2 = x 2 + 2x + 6 b. ( a + b) 2 = a 2 + 2ab + b 2 (5a + b) 2 = (5a) 2 + 2(5a)(b) + (b) 2 = 25a 2 + 0ab + b 2 c. ( a + b) 2 = a 2 + 2ab + b 2 ( + c ) 2 = () 2 + 2()( c ) + ( c ) 2 = + 2c + c 6 2a. ( a - b) 2 = a 2-2ab + b 2 ( x - 7) 2 = (x) 2-2(x)(7) + (7) 2 = x 2-4x + 49 b. ( a - b) 2 = a 2-2ab + b 2 (b - 2c) 2 = (b) 2-2(b)(2c) + (2c) 2 = 9b 2-2bc + 4c 2 c. ( a - b) 2 = a 2-2ab + b 2 ( a 2-4 ) 2 = ( a 2 ) 2-2( a 2 ) (4) + (4) 2 = a 4-8a a. (a + b)(a - b) = a 2 - b 2 (x + 8)(x - 8) = (x) 2 - (8) 2 = x Holt McDougal Algebra

21 b. (a + b)(a - b) = a 2 - b 2 ( + 2y 2 )( - 2y 2 ) = () 2 - (2y 2 ) 2 = 9-4y 4 c. (a + b)(a - b) = a 2 - b 2 (9 + r)(9 - r) = (9) 2 - (r) 2 = 8 - r 2 4. Area of : (5 + x)(5 - x) = (5) 2 - (x) 2 = 25 - x 2 Area of : x 2 Total area = area of + area of = ( 25 - x 2 ) + x 2 = 25 + (-x 2 + x 2 ) = 25 The area of the pool is 25. think and discuss. (a + b)(a - b) = a 2 - ab + ab - b 2 = a 2 - b 2 2. product. Special Products of Binomials Perfect-Square Trinomials ( a + b ) 2 = a ab + b 2 ( x + 4) 2 = x x + 6 Exercises guided practice ( a - b ) 2 = a 2-2 ab + b 2 ( x - 4) 2 = x 2-8 x + 6 Difference of Two Squares ( a + b)( a - b) = a 2 - b 2 ( x + 4)( x - 4) = x 2-6. Possible answer: a trinomial that is the result of squaring a binomial. 2. ( a + b) 2 = a 2 + 2ab + b 2 ( x + 7) 2 = (x) 2 + 2(x)(7) + (7) 2 = x 2 + 4x ( a + b) 2 = a 2 + 2ab + b 2 (2 + x) 2 = (2) 2 + 2(2)(x) + (x) 2 = 4 + 4x + x 2 4. ( a + b) 2 = a 2 + 2ab + b 2 ( x + ) 2 = (x) 2 + 2(x)() + () 2 = x 2 + 2x + 5. ( a + b) 2 = a 2 + 2ab + b 2 (2x + 6) 2 = (2x) 2 + 2(2x)(6) + (6) 2 = 4x x ( a + b) 2 = a 2 + 2ab + b 2 (5x + 9) 2 = (5x) 2 + 2(5x)(9) + (9) 2 = 25x x ( a + b) 2 = a 2 + 2ab + b 2 (2a + 7b) 2 = (2a) 2 + 2(2a)(7b) + (7b) 2 = 4a ab + 49b 2 8. ( a - b) 2 = a 2-2ab + b 2 ( x - 6) 2 = (x) 2-2(x)(6) + (6) 2 = x 2-2x ( a - b) 2 = a 2-2ab + b 2 ( x - 2) 2 = (x) 2-2(x)(2) + (2) 2 = x 2-4x ( a - b) 2 = a 2-2ab + b 2 (2x - ) 2 = (2x) 2-2(2x)() + () 2 = 4x 2-4x +. ( a - b) 2 = a 2-2ab + b 2 (8 - x) 2 = (8) 2-2(8)(x) + (x) 2 = 64-6x + x 2 2. ( a - b) 2 = a 2-2ab + b 2 (6p - q) 2 = (6p) 2-2(6p)(q) + (q) 2 = 6p 2-2pq + q 2. ( a - b) 2 = a 2-2ab + b 2 (7a - 2b) 2 = (7a) 2-2(7a)(2b) + (2b) 2 = 49a 2-28ab + 4b 2 4. (a + b)(a - b) = a 2 - b 2 (x + 5)(x - 5) = (x) 2 - (5) 2 = x (a + b)(a - b) = a 2 - b 2 (x + 6)(x - 6) = (x) 2 - (6) 2 = x (a + b)(a - b) = a 2 - b 2 (5x + )(5x - ) = (5x) 2 - () 2 = 25x 2-7. (a + b)(a - b) = a 2 - b 2 (2x 2 + )( 2x 2 - ) = ( 2x 2 ) 2 - () 2 = 4x (a - b)(a + b) = a 2 - b 2 ( 9 - x ) (9 + x ) = (9) 2 - ( x ) 2 = 8 - x 6 9. (a - b)(a + b) = a 2 - b 2 (2x - 5y)(2x + 5y) = (2x) 2 - (5y) 2 = 4x 2-25y Area of big : (x + ) 2 = (x) 2 + 2(x)() + () 2 = x 2 + 6x + 9 Area of small : (x + ) 2 = (x) 2 + 2(x)() + () 2 = x 2 + 2x + Total area = area of big + area of small = (x 2 + 6x + 9) + (x 2 + 2x + ) = (x 2 + x 2 ) + (6x + 2x) + (9 + ) = 2x 2 + 8x + 0 The area of the figure is 2x 2 + 8x + 0. practice and problem solving 2. ( a + b) 2 = a 2 + 2ab + b 2 ( x + ) 2 = (x) 2 + 2(x)() + () 2 = x 2 + 6x Holt McDougal Algebra

22 22. ( a + b) 2 = a 2 + 2ab + b 2 (4 + z) 2 = (4) 2 + 2(4)(z) + (z) 2 = 6 + 8z + z 2 2. ( a + b) 2 = a 2 + 2ab + b 2 (x 2 + y 2 ) 2 = ( x 2 ) 2 + 2( x 2 ) (y 2 ) + (y 2 ) 2 = x 4 + 2x 2 y 2 + y ( a + b) 2 = a 2 + 2ab + b 2 (p + 2q ) 2 = (p) 2 + 2(p)(2q ) + (2q ) 2 = p 2 + 4pq + 4q ( a + b) 2 = a 2 + 2ab + b 2 (2 + x) 2 = (2) 2 + 2(2)(x) + (x) 2 = 4 + 2x + 9x ( a + b) 2 = a 2 + 2ab + b 2 (r 2 + 5t) 2 = ( r 2 ) 2 + 2( r 2 ) (5t) + (5t) 2 = r 4 + 0r 2 t + 25t ( a - b) 2 = a 2-2ab + b 2 ( s 2-7 ) 2 = ( s 2 ) 2-2( s 2 ) (7) + (7) 2 = s 4-4s ( a - b) 2 = a 2-2ab + b 2 ( 2c - d ) 2 = (2c) 2-2(2c)( d ) + ( d ) 2 = 4c 2-4cd + d ( a - b) 2 = a 2-2ab + b 2 ( a - 8) 2 = (a) 2-2(a)(8) + (8) 2 = a 2-6a ( a - b) 2 = a 2-2ab + b 2 (5 - w) 2 = (5) 2-2(5)(w) + (w) 2 = 25-0w + w 2. ( a - b) 2 = a 2-2ab + b 2 (x - 4) 2 = (x) 2-2(x)(4) + (4) 2 = 9x 2-24x ( a - b) 2 = a 2-2ab + b 2 ( - x 2 ) 2 = () 2-2()( x 2 ) + ( x 2 ) 2 = - 2x 2 + x 4. (a - b)(a + b) = a 2 - b 2 (a - 0)(a + 0) = (a) 2 - (0) 2 = a (a + b)(a - b) = a 2 - b 2 (y + 4)(y - 4) = (y) 2 - (4) 2 = y (a + b)(a - b) = a 2 - b 2 (7x + )(7x - ) = (7x) 2 - () 2 = 49x (a - b)(a + b) = a 2 - b 2 ( x 2-2 ) (x 2 + 2) = ( x 2 ) 2 - (2) 2 = x (a + b)(a - b) = a 2 - b 2 (5a 2 + 9)( 5a 2-9 ) = ( 5a 2 ) 2 - (9) 2 = 25a (a + b)(a - b) = a 2 - b 2 (x + y 2 )(x 2 - y 2 ) = ( x ) 2 - (y 2 ) 2 9. A = πr 2 = π(x + 4) 2 = x 6 - y 4 = π((x) 2 + 2(x)(4) + (4) 2 ) = π(x 2 + 8x + 6) = π( x 2 ) + π(8x) + π(6) = πx 2 + 8πx + 6π The area of the puzzle is πx 2 + 8πx + 6π. 40a. x > 2; values less than or equal to 2 cause the width of the rectangle to be zero or negative, which does not make sense. b. Area of : (x - ) 2 = (x) 2-2(x)() + () 2 = x 2-2x + Area of : x(x - 2) = x(x) + x(-2) = x 2-2x Since x 2-2x + > x 2-2x, the square has the greater area. c. Difference = area of - area of = (x 2-2x + ) - ( x 2-2x ) = (x 2-2x + ) + (-x 2 + 2x) = ( x 2 - x 2 ) + (-2x + 2x) + = The difference in area is square unit. 4. ( a + b) 2 = a 2 + 2ab + b 2 ( x + y) 2 = (x) 2 + 2(x)(y) + (y) 2 = x 2 + 2xy + y ( a - b) 2 = a 2-2ab + b 2 ( x - y) 2 = (x) 2-2(x)(y) + (y) 2 = x 2-2xy + y 2 4. (a + b)(a - b) = a 2 - b 2 (x 2 + 4)( x 2-4 ) = ( x 2 ) 2 - (4) 2 = x ( a + b) 2 = a 2 + 2ab + b 2 (x 2 + 4) 2 = ( x 2 ) 2 + 2( x 2 ) (4) + (4) 2 = x 4 + 8x ( a - b) 2 = a 2-2ab + b 2 ( x 2-4 ) 2 = ( x 2 ) 2-2( x 2 ) (4) + (4) 2 = x 4-8x ( a - b) 2 = a 2-2ab + b 2 ( - x) 2 = () 2-2()(x) + (x) 2 = - 2x + x 2 26 Holt McDougal Algebra

23 47. ( a + b) 2 = a 2 + 2ab + b 2 ( + x) 2 = () 2 + 2()(x) + (x) 2 = + 2x + x 2 6. Possible answer: The square of a difference is not the same as a difference of squares; a 2-2ab + b 2. 64a. 48. (a - b)(a + b) = a 2 - b 2 ( - x)( + x) = () 2 - (x) 2 = - x 2 x + x (a - b)(a - b) = a 2-2ab + b 2 ( x - a ) ( x - a ) = ( x ) 2-2( x ) ( a ) + ( a ) 2 = x 6-2x a + a ( a + b)(a + b) = a 2 + 2ab + b 2 (5 + n)(5 + n) = (5) 2 + 2(5)(n) + (n) 2 = n + n 2 5. (a - b)(a + b) = a 2 - b 2 (6a - 5b)(6a + 5b) = (6a) 2 - (5b) 2 = 6a 2-25b (a - b)(a - b) = a 2-2ab + b 2 ( r - 4t 4 ) ( r - 4t 4 ) = (r) 2-2(r) ( 4t 4 ) + ( 4t 4 ) 2 = r 2-8rt 4 + 6t 8 a b ( a - b) 2 a 2-2ab + b 2 4 ( - 4) 2 = 9 () 2-2()(4) + (4) 2 = (2-4) 2 = 4 (2) 2-2(2)(4) + (4) 2 = ( - 2) 2 = () 2-2()(2) + (2) 2 = a b ( a + b) 2 a 2 + 2ab + b ( + 4) 2 = 25 () 2 + 2()(4) + (4) 2 = (2 + 5) 2 = 49 (2) 2 + 2(2)(5) + (5) 2 = ( + 0) 2 = 9 () 2 + 2()(0) + (0) 2 = 9 a b (a + b)(a - b) a 2 - b ( + 4)( - 4) = -5 () 2 - (4) 2 = (2 + )(2 - ) = -5 (2) 2 - () 2 = ( + 2)( - 2) = 5 () 2 - (2) 2 = 5 6. a b = ( a + b) 2 - (a - b) = (5 + 24)2 - (5-24) 2 4 = (59)2 - () = 4 = _ 60 4 = Notice that: ( a - b) 2 = a 2-2ab - b 2 = 6x 2-24x + c Therefore, a 2 = 6x 2 = (4x) 2. So a = ±4x. Therefore, -24x = -2ab = -2(±4x)b = 8xb. -24x = 8xb _ -24x = 8xb _ 8x 8x ± = b So c = b 2 = (±) 2 = 9. b. A = lw = (x + )(x - ) = (x) 2 - () 2 = x 2-9 The area is represented by x 2-9. c. P = 2l + 2w 48 = 2(x + ) + 2(x - ) 48 = 2(x) + 2() + 2(x) + 2(-) 48 = 2x x = 2x + 2x = 4x 48 4 = 4x 4 2 = x A = x 2-9 = (2) 2-9 = 44-9 = 5 The area of the region is 5 ft For ax 2-49 to be a perfect square, ax 2 needs to be a perfect square. Therefore, a must be a perfect square. So all the possible values of a are all the perfect squares from to 00;, 4, 9, 6, 25, 6, 49, 64, 8, When one binomial is in the form a + b and the other is in the form a - b; (x + 2)(x - 2) = x 2-4. test prep 67. B (a - b)(a - b) = a 2-2ab + b 2 (5x - 6y)(5x - 6y) = (5x) 2-2(5x)(6y) + (6y) 2 = 25x 2-60xy + 6y J; The 25x 2 region means ±5x is squared. The 4 region means ±2 is squared. The two 0x regions mean that the product of ±5x and ±2 is positive, so the terms have the same sign. Therefore, it must be J. 69. D; If a = 0, then b = 2 from the first equation. Notice that (0) 2 - (2) 2 = 00-4 = 96, so a = 0, b = 2 is a solution to both equations. Therefore, a = H; Notice that (r + s) 2 = r 2 + 2rs + s 2 = 64. Since rs = 5, r 2 + 2(5) + s 2 = 64, or r 2 + s 2 = Holt McDougal Algebra

24 challenge and extend 7. (x + 4)(x + 4)(x - 4) = (( x) 2 + 2(x)(4) + (4) 2 )( x - 4) = (x 2 + 8x + 6)( x - 4) = (x - 4)(x 2 + 8x + 6) = x(x 2 + 8x + 6) - 4(x 2 + 8x + 6) = x( x 2 ) + x(8x) + x(6) - 4 ( x 2 ) - 4(8x) - 4(6) = x + 8x 2 + 6x - 4x 2-2x - 64 = x + 4x 2-6x (x + 4)(x - 4)(x - 4) = (( x) 2 - (4) 2 )( x - 4) = ( x 2-6 ) ( x - 4) = x 2 ( x) + x 2 ( -4) - 6(x) - 6(-4) = x - 4x 2-6x Let x 2 + bx + c = x 2 + bx + (± Ç c ) 2 since c = (± Ç c ) 2. x 2 + bx + (± Ç c ) 2 = (x± Ç c )(x± Ç c ) because the trinomial is a perfect square. (x± c Ç )(x± Ç c ) = x 2 ± 2 Ç c x + (± Ç c ) 2 by multiplication. Make the coefficients of x: b = ±2 Ç c. 74. Rewrite 27 as and 9 as = (2 + 4)(2-4) = (2) 2 - (4) 2 = = 5 ready to go on? Section B Quiz. 4r 2 + 2r 6 - r 2r 6 + 4r 2 - r The leading coefficient is y y + 2y -8y + y 2 + 2y + 7 The leading coefficient is t - 4t + t 4 t 4-2t - 4t The leading coefficient is. 4. n + + n 2 n 2 + n + The leading coefficient is x x + 2 The leading coefficient is. 6. -a a 7 + a a 7 - a 2 + a + 6 The leading coefficient is. 7. Degree: Terms: 2x + 5x - 4 is a cubic trinomial. 8. Degree: 2 Terms: 5b 2 is a quadratic monomial. 9. Degree: 4 Terms: 4 6p 2 + p - p 4 + 2p is a quartic polynomial. 0. Degree: 2 Terms: x x is a quadratic trinomial.. Degree: 7 Terms: 4-2x x - 2x 7 is a 7th-degree polynomial. 2. Degree: 4 Terms: 4 5-6b 2 + b - 4b 4 is a quartic polynomial.. C(x) = x - 5x + 4 C(900) = (900) - 5(900) + 4 = 729,000,000 -, = 728,986,54 The cost to manufacture 900 units is $728,986, (0m + 4m 2 ) + (7m 2 + m) = 0m + (4m 2 + 7m 2 ) + m = 0m + m 2 + m 5. ( t 2-2t ) + (9t 2 + 4t - 6) = (t 2 + 9t 2 ) + (-2t + 4t) + (-6) = 2t 2 + 2t ( 2d 6 - d 2 ) + (2d 4 + ) = 2d 6 + 2d 4 - d (6y + 4y 2 ) - (2y 2 + y) = (6y + 4y 2 ) + (-2y 2 - y) = 6y + (4y 2-2y 2 ) + (-y) = 6y + 2y 2 - y 8. ( 7n 2 - n ) - (5n 2 + 5n) = ( 7n 2 - n ) + ( -5n 2-5n ) = ( 7n 2-5n 2 ) + (-n - 5n) = 2n 2-8n 9. ( b 2-0 ) - (-5b + 4b) = ( b 2-0 ) + ( 5b - 4b ) = 5b + b 2-4b P = (2s + 4) + (4s 2 + ) + (5s) = 2s + 4s 2 + 5s + (4 + ) = 2s + 4s 2 + 5s h 5h 5 = (2 5)( h h 5 ) = 0h ab(5a + a 2 b) = 2ab( 5a ) + 2ab ( a 2 b ) 22. ( s 8 t 4 ) ( -6st ) = (-6)( s 8 s ) ( t 4 t ) = -6s 9 t 7 = (2 5)( a a ) ( b) + (2 ) ( a a 2 ) ( b b) = 0a 4 b + 6a b (k + 5) 2 = (k + 5)(k + 5) = k(k) + k(5) + 5(k) + 5(5) = 9k 2 + 5k + 5k + 25 = 9k 2 + 0k (2x + y)(4x 2 + y) = 2x ( 4x 2 ) + 2x ( y) + y ( 4x 2 ) + y(y) = 8x 5 + 2x y + 2x 2 y + y 2 28 Holt McDougal Algebra