30. 2 x5 + 3 x; quintic binomial 31. a. V = 10pr 2. b. V = 3pr 3

Transcription

1 Answers for Lesson 6- Answers for Lesson 6-. 0x + 5; linear binomial. -x + 5; linear binomial. m + 7m - ; quadratic trinomial 4. x 4 - x + x; quartic trinomial 5. p - p; quadratic binomial 6. a + 5a + ; cubic trinomial 7. -x 5 ; quintic monomial 8. x 4 + ; quartic binomial 9. 5x ; cubic monomial 0. -x ; cubic monomial. 5x + 4x + 8; quadratic trinomial. -x 4 + x ; quartic binomial. = x + 4. = x - 5. =.5x + x - x + 6. =-x - 0x a. males: = x + 0.5x females: = x + 0.x b. males: = x x + 0.9x females: = x x x c. For males, the models offer similar fit. For females, the cubic model is a better fit. 8. = x - x ; = x - 0x ; 0 0. =-0.5x + 0x ; 4.5. = x +.069x - 7.9x ; = x x x ; 6.4. = 0.000x x x +.; Check students work. 5. x + 4x; cubic binomial 6. -4a 4 + a + a ; quartic trinomial 7. 7; constant monomial 8. 6x ; quadratic monomial 9. x 4 + x ; quartic binomial 0. x5 + x; quintic binomial. a. V = 0pr b. V = pr c. V = pr + 0pr. Answers ma var. Sample: Cubic functions represent curvature in the data. Because of their turning points the can be unreliable for extrapolation.. -c + 6; binomial 4. -9d - ; binomial 5. 6x - x - 5; trinomial 6. x - 6x + 7; trinomial 7. a + 4b; binomial 8. -; monomial 9. 8x - 6; binomial 40. -a + ; binomial Algebra Chapter 6 8 Algebra Chapter 6 9 Answers for Lesson 6- Answers for Lesson 6-4. x + 9x + 5x + 7; polnomial of 4 terms 4. -4x 4 - x + 5x - 54; polnomial of 4 terms 4. 80x - 09x + 7x - 75; polnomial of 4 terms 44. x - x + 8x - 7; polnomial of 4 terms 59. a. = x = x x +.690x = x x x x b a + ab - 8; trinomial 46. 8x + x ; binomial 47. 0x - 0x ; binomial 48. a - 5a - a + 5; polnomial of 4 terms 49. b - 6b + 9b; trinomial 50. x - 6x + x - 8; polnomial of 4 terms 5. x 4 + x + ; trinomial 5. 8x + 60x + 50x + 6; polnomial of 4 terms 5. a - a b - b a + b ; polnomial of 4 terms 54. a 4-4a + 6a - 4a + ; polnomial of 5 terms 55. s + 6s + 68s - ; polnomial of 4 terms 56. x + x - x - ; trinomial 57. 8c - 6c + ; trinomial 58. s 4 - t s + t 4 ; trinomial Answers ma var. Sample: The quartic model fits best. c. For sample in part (b), Btu cm 6. a. up 4 units b. = 4x is more narrow. c. = x Algebra Chapter 6 40 Algebra Chapter 6 4

2 Answers for Lesson 6- Answers for Lesson 6-. x + x - 6. x + x + 47x = x - 8x + 07x - 0. x - 7x + 5x x + 4x + 4x. = x + x - x 5. x + 0x + 5x 6. x - x. = x + 9x + 5x = x - 9x + 7x x(x - 6)(x + 6) 8. x(x - )(x + ) 5. = x + x - x - 6. = x + 6x + x x(x - x + ) 0. x(x + 5)(x + ) 7. = x - x 8. = x 7 - x - x. x(x + 4). x(x - 9)(x + ) 9. - (mult. ) 0. 0, (mult. ). about 4., -.4, 0, -5, 4. about 5.0, -6.9,, 6, 8 5. a. h = x, / = 6 - x, c. w = - x b. V = x(6 - x)( - x) 94 in.,.6 in. 6., - 7., -9 O x 8. 0, -5, ,, 40 4 O 4 x 80 Maximum X=.6967 Y= O x x O. -, 0,. -, 0,. 4 (mult. ) 4., (mult. ) 5. -, (mult. ) 6. - (mult. ),, 7. x blocks, 5 x blocks, x blocks, unit blocks 8. a. V = x + 5x + x + ; x + 7x + 7x + b. V = 8x + 4x V = x - 7x 40. a. h = x + ; w = x b. 4 O x 4 8 0, -, ; where the volume is zero c. 0, x, 0. -,, O x d. about 4.06 ft 4. =-x + 9x - x - 4. = 5x 4 - x - 50x + 64x Algebra Chapter 6 4 Algebra Chapter 6 4 Answers for Lesson 6- Answers for Lesson 6-4. = x(x - 8)(x - ) 44. =-x(x + 5)(x - 4). x - 8. x = x (x + 4)(x - ) 46. = xqx RQx R. x + 4x +, R 5 4. x + 5x about 0.5, -7.;, 4, 6 5. x - 7x x -, R about 0.9, -6.9, -.4; 0, -, -, 7. x - 0, R x + 4x about -.98, -6.7; none, -; -, 0 9. no 0. es 5 5. Answers ma var. Samples are given.. es. no 5. = x - x - 0x. x + 4x + 4. x - x + 5. = x - x + 47x x - x + 7, R x + x = x 4-4x - 7x + x , 5 (mult. ) (mult. ), - (mult. ) 56. 0, 6, Answers ma var. Sample: Write the polnomial in standard form. The constant term is the value of the -intercept. 58. ft 59. Answers ma var. Sample: = x 4 - x, and zeros are 0, Answers ma var. Sample: The linear factors can be determined b examining the x-intercepts of the graph. 6. x + a 6. a. A =-x + x + 4x 7 b. 68 square units 6. Answers ma var. Sample: = (x - )(x + )(x - i)(x + i); = x a. Answers ma var. Sample: translation to the right 4 units b. No; the second graph is not the result of a horizontal translation. c. Answers ma var. Sample: rotation of 80 about the origin 7. x - x x + 9x - 9, R x +, R 4 0. x + 8x -. x - x x -, R -4. = (x + )(x + )(x - ) 4. = (x + )(x - 4)(x - ) 5. / = x + and h = x P(a) = 0; x - a is a factor of P(x). 5. x - is not a factor of x - x - x because it does not divide into x - x - x evenl. 6. Answers ma var. Sample: (x + x - 4) 4 (x - ) 7. x + 4x x - x + x - 5, R x 4 - x + x - x x x - x + 4. no 4. es 44. es 45. no 46. no Algebra Chapter 6 44 Algebra Chapter 6 45

3 Answers for Lesson es 48. es 49. es 50. no Answers for Lesson ,, 5. -, 0, 5. no. 0, 4. 0, 8 5. x - x , -, , -.5, 5. x - x - x + 4, R , -0.5, , 0, 54. x - x - x , % 55. x - 4x + x. about 5.78 ft 6.78 ft.78 ft 56. a. x + b. x + x + c. x + x + x + d. (x - )(x 4 + x + x + x + ) 57. a. x - x + b. x 4 - x + x - x + c. x 6 - x 5 + x 4 - x + x - x + d. (x + )(x 8 - x 7 + x 6 - x 5 + x 4 - x + x - x + ) 58. B dividing it b a polnomial of degree, ou are reducing the degree-n polnomial b one, to n -. The remainder will be constant because it is not divisible b the variable. 59. x + i 60. Yes; the graph could rise to the right and fall to the left or it could fall to the right and rise to the left. O x. (x + 4)(x - 4x + 6). (x - 0)(x + 0x + 00) 4. (5x - )(5x + 5x + 9) 4 i! 5 4 5i! 5., 6. -4, 4 i! 7. 5, 4 i! 4 i! 4 i! 8. -, 9., , 4. (x - 7)(x - )(x + ). (x + 0)(x - ). (x - )(x - )(x + ) 4. (x - )(x + )(x - )(x + ) 5. (x - )(x + )(x + ) 6. (x - )(x + )(x - ) 7. 4, , 4i 0. 4i, 4!. 4!, 4i!6. 4i!5, 4i!..4, -, , , -,, 6..7, ,.54, ,.7, 4.7 Algebra Chapter 6 46 Algebra Chapter 6 47 Answers for Lesson 6-4 Answers for Lesson , 0, (n - )(n)(n + ) = 0; 5, 6, 7 4. D 6 4 i! 4 4 i! , 5 4., 44. 4!,4i! , 4i! 69. a. Answers ma var. Sample: x 4-9 = 0, 4!, 4i! b. No; two of the roots are imaginar. 70. Answers ma var. Sample: The pink block has volume a (a - ), the orange block has volume 9(a - ), the blue block has volume a(a - ), and the purple block has volume 7. Thus a - 7 = a (a - ) + a(a - ) + 9(a - ) = (a + a + 9)(a - ) i, 4i! 47. 0, 4, !0, 4i! , 4! a. 0 b. 8 and 50. 4, - 4 i! 5. 0, 4! 5. -,0,4 5. -,, 4i! , -, 55. -,, 56. 0,, 57. 0, 0,, Å, 4i 59. 4!, 4i 60. Check students work. 6. V = x (4x - ), 4 in. b 4 in. b 6 in. 6. x = length, V = x(x - )(x - ), 5 meters ,; = (x + 5)(x - ) 64. 4, 4; = (x - )(x + )(x - )(x + ) 65. -,, ; = (x + )(x - ) 66. -,, ; = (x + )(x - )(x - ) , -, ; = (x + 4)(x + )(x - ) 68. A cubic can onl have zeros. Algebra Chapter 6 48 Algebra Chapter 6 49

4 Answers for Lesson , 4;. 4, 4, 4, 46;, -, -. 4, 4, 44; , 4, 4, 44, 48; no rational roots 5. 4, 4, 44, 48, 46; , 4, 45, 45; no rational roots 7., 4i!5 8. 5, 4i!7 7 4! 9. -,, 0. -5, 4!7. 4, 4., -,. -!5,! !6,! 5. +!0,-! 6. - i,5i 7. - i, -6i i,- 7i 9. x - x + 9x - 9 = 0 0. x + x - 8x + 0 = 0. x - x + 6x - = 0. x - x - 8x + 0 = 0. x - 6x + 4x - 4 = 0 4. x - x + = , 4 6, 4 4, 4, 4, 4, 4 4,4, 4, 4, 4, 46;,, , 4 5, 4 5, 4, 4 4 5,4, 4, 4 5, 44, 45, 40, 40;, 5, , 4 6, 4, 4, 4 7 6,4, 4, 4, 4 7, 47, 4, 4;, 7,, , 4 5 4, 4, 4 4, 4 8, 4 5 8, 4 8, 4 5 4, 4 5,4, 4, 4 5 8, 4 5, 45, 5 4, 45;,, 9. x 4-6x + 4x - 4x + 40 = 0 0. x 4 - x - x + 6x - 6 = 0. x 4-6x + x + 0x - 5 = 0 Answers for Lesson 6-5. Never true; 5 is not a factor of 8, so b the Rational Root Theorem, 5 is not a root of the equation.. Sometimes true; since - is a factor of 8, - is a possible root of the equation. 4. Alwas true; use the Rational Root Theorem with p = a and q =. 5. Sometimes true; since!5 and!5 are conjugates, the can be roots of a polnomial equation with integer coefficients. 6. Never true; since + i and - - i are not conjugates, the cannot be the onl imaginar roots of a polnomial equation with integer roots. If their conjugates were also roots, there would be four roots and the equation would have to be of fourth degree. 7. D 8. If i is a root, then so is -i. 9. Answers ma var. Sample: x 4 - x - = 0; roots are and 4i. 40. a. real, imaginar; 4 imaginar; 4 real b. 5 real; real, imaginar; 4 imaginar, real c. Answers ma var. Sample: It has an odd number of real solutions, but it must have at least one real solution. 4. Answers ma var. Sample: You cannot use the Irrational Root Theorem unless the equation has rational coefficients. 4. x + (- + i)x + - 8i = 0 4. a c. Answers ma var. Sample: a. x - -! = 0 b. x - ( +!)x + ( +!) = 0 c. - 4! Algebra Chapter 6 50 Algebra Chapter 6 5 Answers for Lesson 6-6. complex roots; number of real roots: or possible rational roots: 4. complex roots; number of real roots: 0 or possible rational roots: 4, 4 7, 4, complex roots; number of real roots: 0,, or 4 possible rational roots: complex roots; number of real roots:,, or 5 possible rational roots: 4, 4, 4 5, 45 Answers for Lesson complex roots; number of real roots: 0,, 4, or 6 possible rational roots: 4 4, 4, 4 4,4, 4, 4, 4, 44, 46, 48, 4, 44 4 i. 4, 4i. -, 4!5. -6, 4 i! 4. 4, - 4 i 5., 4i!5 6. 5, 6 7. Answers ma var. Sample: = x 4 + x , complex roots; number of real roots:,, 5, or 7 possible rational roots: 4, 4 6. complex root number of real roots: possible rational roots: 4 4, 4, 4, 4, 44, complex roots; number of real roots: 0,, 4, or 6 possible rational roots: 4, 4, 4 7, complex roots; number of real roots: 0,, 4, 6, 8, or 0 possible rational roots: 4 4 i!7 9. -, 4 0., 4i 4 i!. 4,., 4!. 4, 4! 4. 4, 4i 4!5 5. 0, 6. -6, 4i 7. 4 complex roots; number of real roots: 0,, or 4 possible rational roots: 4, 4, 4, 4, 4, complex roots; number of real roots:,, or 5 possible rational roots: 4, 4, 4, 46, 49, If ou have no constant, then all terms have an x that can be factored out. The resulting expression will have a constant that can be used in the Rational Root Theorem.. Yes; for example, x - x + 5 = 0 has roots 0.5 and complex roots; number of real roots: or possible rational roots: 4, 4, 4, 4 4, 4, 4, 44, 46, 4 Algebra Chapter 6 5 Algebra Chapter 6 5

5 Answers for Lesson ,68,800. 6,7,00, , a. 4 b , ,897,86, ,5,47, true because of the Comm. Prop. of Add. 4. true because of the Assoc. Prop. of Mult. 5. False; answers ma var. Sample: ( + )! = 0 and! +! = 8 6. False; answers ma var. Sample: ( )! = 70 and!! = 7. False; answers ma var. Sample: (!)! = 70 and (!) = 6 8. False; answers ma var. Sample: (!) = 6 and (!) = 9 Answers for Lesson < , permutation 47. permutation 48. combination 49. combination 50. 0,79, , a. 56 b. 56 c. Answers ma var. Sample: Each time ou choose of the 8 points to use as vertices of a n, the 5 remaining points could be used to form a pentagon Check students work. 68. a. 048 b. Answers ma var. Sample: No, because there are too man possible solutions. 69. a. The graph for = x C x- is identical to the graph for = x C because C - = C, C - = C, 4 C 4- = 4 C, 5 C 5- = 5 C, etc. b. Answers ma var. Sample: The function is defined onl at discrete whole-number values of x, and not over a smooth range of points as in a continuous function. 70. a. 5 b. 6 c. 7 C = 7!, so 7 C?! = 7!, which is the permutation! 4! 4! formula for 7 P. 9. C was, because order matters Algebra Chapter 6 54 Algebra Chapter 6 55 Answers for Lesson a. All the terms contain the factors and 5. Since multiplication is commutative, 5 = 0 and 0 times an integer ends in zero. b. 4 zeros. a + a b + ab + b. x - x +. a 4 + 4a b + 6a b + 4ab + b 4 7. Answers ma var. Sample: x 5-5x 4 + 0x - 0x + 5x a. 8 people b c. 658,008 d. about 0.0 or.% 5. a 6-6a 5 b + 5a 4 b - 0a b + 5a b 4-6ab 5 + b 6 6. x 7-7x 6 + x 5-5x 4 + 5x 4 - x 5 + 7x x 8 + 8x 7 + 8x x x x 5 + 8x 6 + 8x d 9 + 9d 8 + 6d d 6 + 6d 5 + 6d d + 6d + 9d + 9. x - 9x + 7x a 4 + a b + 54a b + 08ab + 8b 4. x 6 - x x 4-60x + 40x - 9x x 8 - x x 6-584x 5 + 7,90x 4-57,44x + 4,688x -,07x + 65,56. x 4 + 4x + 6x + 4x w 5 + 5w 4 + 0w + 0w + 5w + 5. s - st + t 6. x 6-6x 5 + 5x 4-0x + 5x - 6x + 7. x 4-4x + 6x - 4x p 7 + 7p 6 q + p 5 q + 5p 4 q + 5p q 4 + p q 5 + 7pq 6 + q7 9. x 5-5x x - 70x + 405x x + x - x Algebra Chapter 6 56 Algebra Chapter 6 57

6 . a. about 5% b. about % c. about %. about 66%. x 7 + 7x 6 + x 5 + 5x 4 + 5x 4 + x 5 + 7x x 8-40x x x 5 + 4,750x ,000x ,500x 6-65,000x , x 4-08x + 54x - x x 5-0x x - 640x + 80x ,649-0,684x + 44,060x - 54,880x +,760x 4-44x x x + 6x + 54x x 4 + x x 6-6x 4 + x - 8. x 6 + 6x 5 + 5x 4 + 0x + 5x + 6x +. x 6-6x 5 + 5x 4-0x + 5x - 6x +. x 5 + 0x x + 80x + 80x + 4. x 5-0x x - 80x + 80x x x + 6x + 6x x + 5x + 5x x 6-84x x 4-80x + 960x 4-84x x 4 + 6x + 6x + 96x x x x x x x 5 + x x 6-8x 5 + 5x 4-540x + 5x 4-458x x + 5x + 75x a. about % 44. a. 6 b. about 6% c. about 6% b C 4 x , 7r 6 s x x 50. 7x x 0 5. x ,680x ,98x ,568x x Answers ma var. Sample: Since one of the terms is negative and it is alternatel raised to odd and even powers, the term is negative when raised to an odd power and positive when raised to an even power. 59. a. (s + 0.5) b. s +.5s s The exponent of q should be 5 because the exponent of q should be the degree (7) minus the exponent of p. 9. x x x + 40x + 0x Algebra Chapter 6 58 Algebra Chapter , d,d e 6. 6, x 5, -5x , a 5,80a 4 b 64. 8, x 7, -x a. -4 b. ( - i) 4 = - 4i i + = (- +!? i) =- + i! i! = Answers ma var. Sample: A coin is tossed five times with the probabilit of heads on each toss 0.5. Write an expression for the probabilit of exactl heads being tossed. 68. a. (k + )! = (k + )? (k)? (k - )?...? = (k + )[(k)? (k - )?...? ] = (k + )? k! b. The derivation below finds a common denominator for the fractions that represent n C k and n C k+, and then uses algebra to show that n C k + n C k+ = n+ C k+.in addition, the identit from part (a) is used three times. n C k + n C k+ = n! k!(n k)! n! (k )!(n k )! = (k )n! (k )!(n k)! (n k)n! (k )!(n k)! = (k n k)n! (n )n! (k )!(n k)! = (k )!(n k)! = (n )! (k )!(n k)! = (n )! (k )!((n ) (k ))! = n+ C k+ c. If ou consider the row of Pascal s Triangle containing just to be row zero, 4 C is 6, the third entr in the fourth row. 4 C is 4, the fourth entr in the fourth row. 5 C is 0, the fourth entr in the fifth row. 4 C + 4 C = = 0 = 5 C Algebra Chapter 6 60