Solutions manual technical mathematics with calculus 5th edition calter textmark

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Solutions Manual Technical Mathematics with Calculus 5th Edition Calter. Full file at https://fratstock.eu/Solutions-Manual-Technical-Mathematics-with-Calculus-5th-Edition-Calter
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Full file at https://fratstock.eu 34Chapter 2: Introduction to Algebra CHAPTER 2 Exercise 1 • Algebraic Expressions Mathematical Expressions Which of the following mathematical expressions are also algebraic expressions? [See Example 1] 1. x + 2y: algebraic expression; contains only algebraic symbols and operations 2.y – log x: not an algebraic expression; it is transcendental3.3 sin x: not an algebraic expression; it is transcendental4.x2 – z3: algebraic expression; contains only algebraic symbols and operationsAlgebraic Expressions Which of the following algebraic expressions are literal expressions? [See Examples 4 and 5] 5. 5xy – 2x: not a literal expression; letters at end of alphabet generally represent variables 6.ax + by: literal expression; letters at beginning of alphabet generally represent constants7.2az – 3bx: literal expression; letters at beginning of alphabet generally represent constants8.4x2 + 4y2: not a literal expression; letters at end of alphabet generally represent variablesTerms How many terms are there in each expression? [See Examples 8 and 9] 9. x3 − 2x: 2 terms 10. (5y + y2) – 5: 2 terms; (5y + y2) is considered one term because of the symbol of grouping 11. (z – 9) (z + 4): 1 term; terms (except within symbols of grouping) are separated by plus and/or minus signs 12. 5(1 + x) + 3(4 – 3x): 2 terms; 5(1 + x) is the first term, 3(4 – 3x) is the second term Factors Write the factors of each expression. [See Examples 10 and 11] 13. 3ax: 3, a, x 14. 9xyz: 3 (3 · 3 = 9), x, y, z 15. 7x2y3: 7, x (x · x = x2), y (y · y · y = y3) 16. 6a2bx: 2, 3 (2 · 3 = 6), a (a · a = a2), b, x Coefficient Write the coefficient of each term. [See Examples 12 and 13] 17. 6x2: 6 is the coefficient 18. x: 1 is the coefficient 19. −x: −1 is the coefficient 20. 3cx3: 3c (letters at the beginning of the alphabet are considered constants) 21.ay 2 a 2 a = ( y ) : is the coefficient 2 2 2Full file at https://fratstock.eu Chapter 222. (1/3)(x + 1) (b) = (b/3) (x + 1): b/3 is the coefficient 23. (c – 2) 3x5: 3(c – 2) is the coefficient 24. 2ax5: 2a is the coefficient Degree State the degree of each term. [See Examples 14 and 15] 25. 3x: first degree (the exponent on x is 1) 26. 4y2: second degree 27. 3xy: second degree (the degree of x is 1, the degree of y is 1; 1 + 1 = 2) 28. 5x2y3: fifth degree (2 + 3 = 5) State the degree of each expression. 29. 3x + 4: first degree 30. 5 – xy: second 31. 3x2 – 2x + 1: second (the highest degree on x is 2) 32. 2xy2 + xy – 4: third degreeExercise 2 • Adding and Subtracting Polynomials Combining Like Terms Combine as indicated and simplify. [See Examples 20, 21, 22, 23, 24 and 25] 1. 8y + 2y ⇒ (8 + 2)y = 10y 2.6x – 8x ⇒ (6 – 8)x = −2x3.38.2a – 17.2a ⇒21.0a4.2.94z + 5.37z ⇒ 8.31z5.5z + 9z – 20z ⇒ (5 + 9 – 20)z = −6z6.9xyz – 2xyz + 7xyz ⇒ 14xyz7.7.39y – 6.62y + 1.94y ⇒ 2.71y8.23.9ab + 54.9ab – 65.1ab ⇒ 13.7ab9.5x + 2x – 8x – x ⇒ −2x10. 9a – 2a +7a – 3a ⇒ 11a 11. 2x2 + 4x2 – 3x2 ⇒ 3x2 12. 5x3 − x3 − 2x3 ⇒ 2x3 13. 7z2 + 6z2 − z2 – 4z2 ⇒ 8z2 14. 3.84z3 – 1.27z3 − 4.32z3 – 7.52z3 ⇒ −9.27z335Full file at https://fratstock.eu 36Chapter 215. 88x + 23y – 17z + 68y – 36x + 39z ⇒ (88 – 36)x +(23 + 68)y – (17 – 39)z ⇒52x + 91y + 22z 16. a – 6b + 2c + a + 6b + n – 2c ⇒ 2a + n 17. 1.95x – 4.38z + 2.83a – 5.21z – 9.27x ⇒ 2.83a – 7.32x – 9.59z 18. 33.9ab – 82.4ac + 29.3ad – 84.2ac + 73.2ab ⇒ (33.9 + 73.2)ab – (82.4 + 84.2)ac + ( 29.3)ad ⇒ 107.1ab – 166.6ac + 29.3ad Combining More Than Two Expressions Combine and simplify. Use either horizontal or vertical addition and subtraction. [See Examples 26, 27 and 28] 19. (4ab + 6bc + 8cd) + (6ab – 3ab) + (4cd – 6bc) ⇒ (4ab + 6ab – 3ab) + (6bc – 6bc) + (8cd + 4cd) ⇒ 7ab + 12cd 20.( 2a + 3b − 1) + ( 2c + d − b ) − ( 3a − 4c + 5 − 6d ) ⇒ ( 2a − 3a ) + ( 3b − b ) + ( 2c + 4c ) + ( d + 6d ) + ( −5 − 1) ⇒ − a + 2b + 6c + 7 d − 621.( 3b − 7 p + 4r ) + ( 3s − 11 p − 19r ) − ( −3 p + r − 10s ) − ( −5b + 2 p − 2r + 4s ) 3b − 7 p + 4r − 11 p − 19r + 3s 3 p − r + 10 s 5b − 2 p + 2r − 4 s 8b − 17 p − 14r + 9 s22.( 2bx + 9 x + 4a ) − ( 4bx − a ) + ( 3b + a ) ⇒23.( −a + 5 x ) + ( a 3 + 3a − 11x 2 + 2a 2 − 3) − ( 7a 2 + 3a3 − 6 x3 + 12 )4a + a + a + 3b + 9 x + 2bx − 4bx ⇒ 6a + 3b + 9 x − 2bx⇒ − a + 5 x + a 3 + 3a − 11x 2 + 2a 2 − 3 − 7 a 2 − 3a 3 + 6 x 3 − 12 ⇒ −2a 3 − 5a 2 + 2a + 6 x 3 − 11x 2 + 5 x − 1524.( 2 y − x + 3 z − 13) − ( 4 x + 2 − 5 y ) + ( 6 z + 8 ) ⇒ 2 y − x + 3 z − 13 − 4 x − 2 + 5 y + 6 z + 8 ⇒ −5 x + 7 y + 9 z − 7Instructions Given Verbally [See Examples 29 and 30] 25. ( a − c + b ) + ( b + c − a ) = ( a − a ) + ( b + b ) + ( c − c ) ⇒ 2b 26.( 6bc + n27.(8b − 10c + 3a − d ) − ( 5a + 7 d − 4b + 6c ) = 8b − 10c + 3a − d − 5a − 7 d + 4b − 6c ⇒ −2a + 12b − 16c − 8d28.( 5 x − 2 xy + 8 y ) − ( 2 xy + 4 y − 3 x ) = 5 x − 2 xy + 8 y − 2 xy − 4 y + 3 x ⇒ 8 x − 4 xy + 4 y29.( 24by52+ 3 p ) + ( −5 x + 3n 2 ) ⇒ −5 x + 6bc + n 2 + 3n 2 + 3 p ⇒ −5 x + 6bc + 4n 2 + 3 p− 14bx 4 ) + ( −72bx5 + 2by 5 − 3bx 4 ) + ( 9bx 4 + 23by 4 − 21by 5 )= 24by 5 + 2by 5 − 21by 5 − 14bx 4 − 3bx 4 + 9bx 4 + 23by 4 − 72bx5 ⇒ 5by 5 − 8bx 4 − 72bx5 + 23by 4 30. ⎣⎡3b ( y 2 − z ) − 6nx 2 ⎤⎦ + ⎡⎣ 2b ( z + y 2 ) + 3nx 2 ⎤⎦ + ⎡⎣ 4b ( − z − y 2 ) − 2nx 2 ⎤⎦ = 3by 2 − 3bz − 6nx 2 + 2bz + 2by 2 + 3nx 2 − 4bz − 4by 2 − 2nx 2⇒ 3by 2 + 2by 2 − 4by 2 − 3bz + 2bz − 4bz − 6nx 2 + 3nx 2 − 2nx 2 ⇒ by 2 − 5bz − 5nx 2 ApplicationsFull file at https://fratstock.eu Chapter 231. 2 ⎣⎡ 2w2 + 3w2 + 6 w2 ⎤⎦ = 2 ⎣⎡11w2 ⎤⎦ ⇒ 22 w2 32. 0.12 x + 0.08 ( 5000 − x ) = 0.12 x + 400 − 0.08 x ⇒ 0.04 x + 400 33. w + l + w + l ⇒ w + w + l + l =2w + 2l 34. π r 2 + 2π rh + π r 2 = 2π r 2 + 2π rh ⇒ 2π r ( r + h ) 35. s2 − s1 = ( 9.76t 2 + 2.95t + 1.94 ) − ( 3.74t 2 + 5.83t + 4.22 ) ⇒ 6.02t 2 − 2.88t − 2.28Exercise 3 • Laws of Exponents Definitions Evaluate each expression. [See Example 32] 1. 33 ⇒ 3 · 3 · 3 = 27 2.(3)5 ⇒ 3 · 3 · 3 · 3 · 3 = 2433.(−2)4 ⇒ (−2) (−2) (−2) (−2) = 164.(−2)5 ⇒ (−2) (−2) (−2) (−2) (−2) = −325.(0.001)3 ⇒ (0.001) (0.001) (0.001) = 1 × 10−96.(−5)3 ⇒ (−5) (−5) (−5) = −125Multiplying Powers Multiply. [See Example 33 and 34] 7. (x4) (x2) ⇒ [(x) (x) (x) (x)] [(x) (x)] ⇒ [(x) (x) (x) (x) (x) (x)] = x6 8.(yb) (y3) ⇒ yb + 39.(a3) (a6) ⇒ a3 + 6 = a9or (x4) (x2) ⇒ x4 + 2 = x610. (105) (109) ⇒ 105 + 9 = 1014 11. (102) (106) ⇒ 102 + 6 = 108 12. (z11) (z2) ⇒ z11 + 2 = z13 Quotients Divide. Write your answers without negative exponents. [See Example 35] a6 a6 a⋅a⋅a⋅a⋅a⋅a a ⋅ a ⋅ a ⋅ a ⋅a⋅a 13. = a 2 or ⇒ a6−4 = a2 ⇒ ⇒ 4 a4 a a⋅a⋅a⋅a a⋅a⋅a⋅a 14.24 ⇒ 24 − 2 = 21 ⇒ 2 2315.y a +1 a +1 − a − 2 ⇒ y ( ) ( ) ⇒ y a +1− a + 2 = y 3 ya−237Full file at https://fratstock.eu 38Chapter 216.106 ⇒ 106 − 2 = 104 10217.10b −1 b −1 − b − 3 ⇒ 10( ) ( ) ⇒ 10b −1− b + 3 = 102 10b − 318.104 ⇒ 104 − 3 = 101 ⇒ 10 10319.b −4 −4 − −5 ⇒ b ( ) = b1 ⇒ b b −520.1 6 1 1 1 x −6 1 x ⇒ ⇒ 6 ÷y⇒ 6 ⋅ = 6 y y x x y x ythis line means “divided by”Power Raised to a Power Simplify. [See Example 36] 21. (a2)4 ⇒ (a · a)4 ⇒ (a · a) (a · a) (a · a) (a · a) ⇒ a · a · a · a · a · a · a · a = a8 22. (25)2 ⇒ 25 · 2 = 210 23. (zc)a ⇒ zc · a = zac 24. (y−1)−3 ⇒ y−1(−3) = y3 25. (ax – 1)3 ⇒ a3(x – 1) = a3x – 3 Product Raised to a Power Raise to the power indicated and remove parentheses. [See Example 37] 26. (ac)3 ⇒ (ac) (ac) (ac) ⇒ (aaaccc) = a3c3 or (ac)3 ⇒ a1 · 3c1 · 3 = a3c3 27. (3a)2 ⇒ 32a2 = 9a2 28. (4a3c2)4 ⇒ 41 · 4a3 · 4c2 · 4 ⇒ 44a12c8 = 256 a12c8 29. (2xyz)5 ⇒ 25x5y5z5 = 32 x5y5z5 Quotient Raised to a Power Raise to the power indicated and remove parentheses. [See Example 38]21⋅3 23 2⋅2⋅2 8 ⎛2⎞ = 30. ⎜ ⎟ ⇒ 1⋅3 ⇒ 3 ⇒ 3 3 3 3 ⋅ 3 ⋅ 3 27 ⎝ ⎠ 3323 8 ⎛ 2⎞ 31. ⎜ − ⎟ ⇒ − 3 = − 5 5 1 25 ⎝ ⎠ 33 ⎛a⎞ a 32. ⎜ ⎟ = 3 ⎝b⎠ b2⎛ 4 x2 ⎞ 41⋅2 x 2⋅2 42 x 4 16 x 4 33. ⎜ 2 ⎟ ⇒ 1⋅2 2⋅2 ⇒ 2 4 = 3 y 3 y 9 y4 ⎝ 3y ⎠or(a2)4 ⇒ a2 · 4 = a8Full file at https://fratstock.eu Chapter 23⎛ 2ab3 ⎞ 23 a 3b 3⋅3 23 a 3b 9 8a 3b9 34. ⎜ 2 ⎟ ⇒ 3 2⋅3 3 ⇒ 3 6 3 = 3c d 3cd 27c 6 d 3 ⎝ 3c d ⎠Zero Exponent Evaluate. [See Example 39] 35.( 2x− 8 x + 32 ) = 1 0236. 108a3c0 ⇒ 108a3(1) = 108a3 37.82 82 ⇒ = 82 0 y 138.c c ⇒ =c y0 1( z )( z ) ⇒ z −n39.z22−nz2−n 2−n⇒ z(2− n)−(2 − n)⇒ z 2−2− n + n ⇒ z 0 = 10⎛ a7 ⎞ ⎛ a 7 ⋅0 ⎞ ⎛ a0 ⎞ ⎛1⎞ 40. 4 ⎜ 4 ⎟ ⇒ 4 ⎜ 4⋅0 ⎟ ⇒ 4 ⎜ 0 ⎟ ⇒ 4 ⎜ ⎟ ⇒ 4 (1) = 4 y y y ⎝1⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠Negative Exponent Write each expression with positive exponents only. [See Example 40] 1 1 41. x −1 ⇒ 1 = x x 42.( −b )−3=−1 b31 −4 4 2 −4 1 1 1 x4 x4 ⎛2⎞ 2 43. ⎜ ⎟ ⇒ −4 ⇒ ⇒ 4÷ 4⇒ 4÷ = 1 x x 2 2 1 16 ⎝x⎠ 4 xa ⎛ 1 ⎞⎛ 1 ⎞ ⎛ a ⎞⎛ 1 ⎞⎛ 1 ⎞ 44. ab −5c −2 ⇒ a ⎜ 5 ⎟⎜ 2 ⎟ ⇒ ⎜ ⎟⎜ 5 ⎟⎜ 2 ⎟ = 5 2 b c 1 b c b c ⎝ ⎠⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠⎛ 1 ⎞⎛ 1 ⎞ 1 −3 6 6 ⎜ ⎟⎜ 9 ⎟ 9 ⎛ 4 x3 ⎞ 4−3 x −9 1 1 ⎛ 1 ⎞ ⎛ 27 y ⎞ 27 y 64 ⎠⎝ x ⎠ ⎝ 64 ⇒ x ⇒ ÷ ⇒⎜ = 45. ⎜ 2 ⎟ ⇒ −3 −6 ⇒ ⎟ 9 9 6 9 ⎟⎜ 1 3 y 64 x 27 y ⎛ 1 ⎞⎛ 1 ⎞ ⎝ 64 x ⎠ ⎝ 1 ⎠ 64 x ⎝ 3y ⎠ ⎜ ⎟⎜ 6 ⎟ 27 y 6 ⎝ 27 ⎠ ⎝ y ⎠ −3⎛ 4 x3 ⎞ An alternative way to solve ⎜ 2 ⎟ based on the definition of the negative exponent ⎝ 3y ⎠ −36 6 ⎛ 4 x3 ⎞ 1 1 1 64 x 9 ⎛ 1 ⎞ ⎛ 27 y ⎞ 27 y ⇒ 3 9 ⇒ ⇒1÷ ⇒ = ⎜ 2⎟ ⇒ ⎜ ⎟ ⎜ ⎟ 3 9 9 9 4 x 64 x 27 y 6 ⎝ 1 ⎠ ⎝ 64 x ⎠ 64 x ⎛ 4 x3 ⎞ ⎝ 3y ⎠ 3 6 6 ⎜ 2⎟ 3 y 27 y ⎝ 3y ⎠39Full file at https://fratstock.eu 40Chapter 2b ⎛ 1 ⎞⎛ b ⎞⎛ 1 ⎞ 46. a −5bc −2 ⇒ ⎜ 5 ⎟⎜ ⎟⎜ 2 ⎟ = 5 2 a 1 c a c ⎝ ⎠⎝ ⎠⎝ ⎠ 3 ⎛ 1 ⎞ ⎛1⎞ 4 47. 4w−4 − 3z −2 ⇒ 4 ⎜ 4 ⎟ − 3 ⎜ 2 ⎟ = 4 − 2 z ⎝w ⎠ ⎝z ⎠ w 1 −4 4 4 4 a −4 ⎛a⎞ ⎛ 1 ⎞⎛ b ⎞ b 48. ⎜ ⎟ ⇒ −4 ⇒ a ⇒ ⎜ 4 ⎟ ⎜ ⎟ = 4 1 b ⎝b⎠ ⎝ a ⎠⎝ 1 ⎠ a b4 −4⎛a⎞ Solving ⎜ ⎟ by the alternative method ⎝b⎠ −44 4 1 1 a4 ⎛a⎞ ⎛1⎞⎛ b ⎞ b ⇒ 4 ⇒ 1÷ 4 ⇒ ⎜ ⎟⎜ 4 ⎟ = 4 ⎜ ⎟ ⇒ 4 a b ⎝b⎠ ⎝1⎠⎝ a ⎠ a ⎛a⎞ 4 ⎜ ⎟ b ⎝b⎠Express without fractions, using negative exponents when needed [See Example 41] 1 49. = a −1 a 50.5 = 5x −3 x351.c2 ⎛ 1 ⎞ ⇒ c 2 ⎜ 3 ⎟ ⇒ c 2 d −3 d3 ⎝d ⎠52.w 4 z −2 ⇒ x −353.⎛ y 2 ⎞⎛ x 4 ⎞ y2 y2 4 2 ⇒ ⇒ ⎜ ⎟⎜ ⎟ = x y 1 x −4 ⎝ 1 ⎠⎝ 1 ⎠ x454.⎛ 1 ⎞⎛ 1 ⎞ ⎜ ⎟⎜ 4 ⎟ c −1d −4 c d ⇒ ⎝ ⎠⎝ ⎠ ⇒ −2 −3 b c ⎛ 1 ⎞⎛ 1 ⎞ ⎜ 2 ⎟⎜ 3 ⎟ ⎝ b ⎠⎝ c ⎠⎛ 1⎞ w4 ⎜ 2 ⎟ ⎝z ⎠⇒ 1 x3w4 4 3 4 3 z 2 ⇒ ⎛ w ⎞⎛ x ⎞ ⇒ w x ⇒ w4 x 3 z −2 ⎜ ⎟ 2 ⎟⎜ 1 z2 ⎝ z ⎠⎝ 1 ⎠ 3 x1 2 3 2 3 cd 4 ⇒ ⎛ 1 ⎞ ⎛ b c ⎞ ⇒ b c ⇒ b 2c3 −1d −4 = b 2c 2 d −4 ⎟ ⎜ 4 ⎟⎜ 1 cd 4 ⎝ cd ⎠ ⎝ 1 ⎠ 2 3 bcApplications 55. (w) (2w) (3w) ⇒(2 · 3) (w · w · w) = 6w356. 16.1(2t)2 ⇒ 16.1(4t2) = 64.4t2 ft ⎛ i2 ⎞ ⎛ i2 ⎞ ⎛ i2 ⎞ ⎛ R ⎞ i2R ⎛i⎞ 57. ⎜ ⎟ R ⇒ ⎜ 2 ⎟ R ⇒ ⎜ ⎟ R ⇒ ⎜ ⎟ ⎜ ⎟ = ⎝ 3⎠ ⎝3 ⎠ ⎝9⎠ ⎝ 9 ⎠⎝ 1 ⎠ 9 258.1 1 1 = + ⇒ R −1 = R1−1 + R2 −1 R R1 R2Full file at https://fratstock.eu Chapter 2Exercise 4 • Product of Two Monomials Multiply the following monomials, and simplify. [See Examples 43, 44, 45, 46, 47, 48, 49 and 50 ] 1. (x) (−y) ⇒ −xy2.(−w) (−z) ⇒ wz3.(a) (−b) (c) ⇒ −abc4.(−x) (−y) (z) ⇒ xyz5.(2.82x) (3.26y) ⇒ (2.82 · 3.26) (xy) ⇒ 9.1932xy = 9.19xy6.(33.5a) (3.72ab) ⇒ (33.5 · 3.72) (a1 + 1b) ⇒ 124.62a2b = 125 a2b (round to 3 significant digits)7.(−8xy) (−2x) ⇒ [(−8) (−2)] (x1 + 1y) ⇒ 16x2y8.(−3ab) (−5ac) ⇒ 15a1 + 1bc = 15a2bc9.a2 (a5) ⇒ a2 + 5 = a7 By calculator:10. xy2(5x2) ⇒ 5x1 + 2y2 = 5x3y2 11. (3.72x2a) (−4.26xa3) ⇒ −15.8472x2 + 1a1 + 3 = −15.8a4x3 By calculator:12. (−6.72ab4) (5.27a2bc) ⇒ −35.4114a1 + 2b4 + 1c = −35.4a3b5c 13. (3.11amx2) (2.94a3) ⇒ 9.1434am + 3x2 = 9.14 am + 3x2 14. (6.82b3x3) (5.16b2x) ⇒ 35.1912b3 + 2x3 + 1 = 35.2b5x4 15. (xmyn) (3x3y2) ⇒ 3xm + 3yn + 2 By calculator:41Full file at https://fratstock.eu 42Chapter 216. (2w3y3) (7wayb) ⇒ 14wa + 3yb + 3Exercise 5 • Product of a Multinomial and a Monomial Multiply the multinomial by the monomial, and simplify. [See Examples 52, 53, 54, 55 and 56] 1. 3(−2 – x) ⇒ 3(−2) – 3(x) = −6 – 3x2.x(b + 2) ⇒ x(b) + x(2) = bx + 2x3.2(a + 3b) ⇒ 2(a) + 2(3b) = 2a + 6b By calculator:4.x(x – 5) ⇒ x(x) – x(5) ⇒ x1 + 1 – 5x = x2 – 5x5.3.83b(b2 + 1.27) ⇒ 3.83b1 + 2 + 3.83b(1.27) ⇒ 3.83b3 + 4.8641b = 3.83b3 + 4.86b6.2.03x(1.27x – 2.36) ⇒ 2.03x1 + 1(1.27) – 2.03x(2.36) ⇒ 2.5781x2 – 4.7905x = 2.58x2 – 4.79x7.3x(−7 – 10x) ⇒ −3x(7) – 3x1 + 1(10) ⇒ −21x – 30x2 = −30x2 − 21x8.b4(b2 + 8) ⇒ b4 + 2 + b4(8) = b6 + 8b49.a2b(2a + b – ab) ⇒ a2 + 1b(2) + a2b1 + 1 – a2 + 1b1 + 1 = 2a3b + a2b2 – a3b2 By calculator:10. −6x3y3(3xy + 5x2y3 − 2xy2) ⇒ −6(3)x3 + 1y3 + 1 + (−6)(5)x3 + 2y3 + 3 − (−6) (2)x3 + 1y3 + 2 = −30x5y6 + 12 x4y5 − 18x4y4 11. 2ab(9a2 + 6ab – 3b2) ⇒ 2(9)a1 + 2b +2(6)a1 + 1 b1 + 1 – 2(3)ab1 + 2 = 18a3b + 12a2b2 – 6ab3 12. −4.27m2(2.83m4 + 6.82m2 – 3.25m3 + 2.47) ⇒ −4.27(2.83)m2 + 4 + (−4.27) (6.82)m2 + 2 − (−4.27) (3.25)m2 + 3 + (−4.27) ( 2.47)m2 ⇒ −12.0841m6 − 29.1214m4 + 13.8775m5 − 10.5469m2 ⇒ −12.1m6 + 13.9m5 − 29.1m4 − 10.5m2Full file at https://fratstock.eu Chapter 24313. −5.16xy(1.23x2y – 5.83xy2 + 4.27x2y2 – 2.94xy) ⇒ −5.16(1.23)x1 + 2y1 + 1 – (−5.16) (5.83)x1 + 1y1 + 2 + (−5.16) (4.27)x1 + 2y1 + 2 – (−5.16) (2.94)x1 + 1y1 + 1 ⇒ −6.3468x3y2 + 30.0828x2y3 – 22.0332x3y3 + 15.1704x2y2 = −6.35x3y2 + 30.1x2y3 – 22.0x3y3 + 15.2x2y2 By calculator:14. 6mn2(5m3n + 4mn2 − 2m2n + mn) ⇒ 6(5)m1 + 3n2 + 1 + 6(4)m1 + 1n2 + 2 − 6(2)m1 + 2n2 + 1 + 6m1 + 1n2 + 1 = 30m4n3 − 12m3n3 + 24m2n4 + 6m2n3Exercise 6 • Product of Two Binomials Multiply the following binomials, and simplify. [See Examples 57, 58, 59 and 60] 1. ( x + y )( x + z ) ⇒ x ( x ) + x ( z ) + y ( x ) + y ( z ) ⇒ x1+1 + xz + xy + yz = x 2 + xz + xy + yz2.( 4a − 3)( a + 2 ) ⇒ 4a ( a ) + 4a ( 2 ) − 3 ( a ) − 3 ( 2 ) ⇒ 4a1+1 + 8a − 3a − 6 = 4a 2 + 5a − 63.( 4m + n ) ( 2m2 − n ) ⇒ 4m ( 2m2 ) + 4m ( −n ) + n ( 2m2 ) + n ( −n ) ⇒ 8m3 − 4mn + 2m 2n − n 2 = 8m3 + 2m2 n − 4mn − n 24.( y + 2 )( y − 2 ) ⇒ y ( y ) − 2 y + 2 y + 2 ( −2 ) = y 2 − 4 (you will factor this later as the difference of squares)5.( 2 x − y )( x + y ) ⇒ 2 x ( x ) + 2 x ( y ) − y ( x ) − y ( y ) = 2 x 2 + xy − y 2 By the distributive property: ( 2 x − y )( x + y ) = 2 x ( x + y ) − y ( x + y ) = 2x ( x) + 2x ( y ) − y ( x) − y ( y ) = 2 x 2 + 2 xy − xy − y 2 = 2 x 2 + xy − y 26.(a7.( 4 xy2− 3b )( a 2 + 5b ) ⇒ a 2 ( a 2 ) + a 2 ( 5b ) − 3b ( a 2 ) − 3b ( 5b ) = a 4 + 2a 2b − 15b 2 2− 3a 3b )( 3 xy 2 + 4a 3b ) ⇒ 4 xy 2 ( 3 xy 2 ) + 4 xy 2 ( 4a3b ) − 3a 3b ( 3 xy 2 ) − 3a 3b ( 4a 3b )⇒ 12 x 2 y 4 + 16 xy 2 a3b − 9 xy 2 a3b − 12a 6b2 = 12 x 2 y 4 + 7a3bxy 2 − 12a 6b2 8.( 2m2− 2n 2 )( 2m 2 + 2n 2 ) ⇒ 2m 2 ( 2m 2 ) + 2m 2 ( 2n 2 ) − 2n 2 ( 2m 2 ) − 2n 2 ( 2n 2 )⇒ 4m 4 + 2 m 2 n 2 − 2 m 2 n 2 − 4n 4 ⇒ 4m 4 − 4n 4 You will see later that a shortcut to multiplying the sum and difference of the same terms is to square the first term, square the second term, and then put a minus (subtraction) sign between the first and second terms.9.( a − 7 x )( 2a + 3 x ) ⇒ a ( 2a ) + a ( 3 x ) − 7 x ( 2a ) − 7 x ( 3x ) ⇒ 2a 2 + 3ax − 14ax − 21x 2 = 2a 2 − 11ax − 21x 2 By calculator:Full file at https://fratstock.eu 44Chapter 210.(3x − z )( 4 x − 3z ) ⇒ 3x ( 4 x ) + 3x ( −3z ) − z ( 4 x ) − z ( −3z ) ⇒ 12 x11.( ax − 5b )( ax + 5b ) ⇒ ax ( ax ) + ax ( 5b ) − 5b ( ax ) − 5b ( 5b ) = a 2 x 2 − 25b 22222222− 9 xz 2 − 4 xz 2 + 3z 4 = 12 x 2 − 13xz 2 + 3z 4By the distributive property: ( ax − 5b )( ax + 5b ) = ax ( ax + 5b ) − 5b ( ax + 5b ) = a 2 x 2 + 5abx − 5abx − 25b 2 = a 2 x 2 − 25b 2 12. ( 5 y + 3 z )( 5 y − 3z ) ⇒ 5 y 2 ( 5 y 2 ) + 5 y 2 ( −3z ) + 3z ( 5 y 2 ) + 3 z ( −3 z ) ⇒ 25 y 4 − 9 z 2 213.2( 2.93x − 1.11y )( x + y ) ⇒ 2.93x ( x ) + 2.93 x ( y ) − 1.11 y ( x ) − 1.11 y ( y ) ⇒ 2.93x 2 + 2.93 xy − 1.11xy − 1.11y 2 ⇒ 2.93x 2 + 1.82 xy − 1.11y 214.( 2.84a2− 3.82b )( a 2 + 5.11b ) ⇒ 2.84a 2 ( a 2 ) + 2.84a 2 ( 5.11b ) − 3.82b ( a 2 ) − 3.82b ( 5.11b )⇒ 2.84a 4 + 14.5124 a 2b − 3.82 a 2b − 19.5202b 2 ⇒ 2.84 a 4 + 10.6924 a 2b − 19.5202b 2 = 2.84a 4 + 10.7 a 2b − 19.5b 215.( 4.03 y2− 3.92a 3b )( 3.26 y 2 + 4.73a 3b ) ⇒ 4.03 y 2 ( 3.26 y 2 ) + 4.03 y 2 ( 4.73a 3b ) − 3.92a 3b ( 3.26 y 2 ) − 3.92a 3b ( 4.73a 3b )⇒ 13.1378 y 4 + 19.0619a3by 2 − 12.7792a3by 2 − 18.5416a 6b 2 ⇒ 13.1378 y 4 + 6.2827 a3by 2 − 18.5416a 6b2 = 13.1y 4 + 6.28a3by 2 − 18.5a 6b 2 16.( 2.83m2− 2.12n 2 )( 2.83m 2 + 2.12n 2 ) ⇒ 2.83m 2 ( 2.83m 2 ) + 2.83m 2 ( 2.12n 2 ) − 2.12n 2 ( 2.83m 2 ) − 2.12n 2 ( 2.12n 2 )⇒ 8.0089m 4 + 5.999m 2 n 2 − 5.999m 2 n 2 − 4.4944n 4 ⇒ 8.0089 m 4 − 4.4944n 4 = 8.01m 4 − 4.49n 4Applications 17. Area of rectangle = length × width ⇒ ( L + 2 )(W − 3) ⇒ L (W ) + L ( −3) + 2 (W ) + 2 ( −3) = LW − 3L + 2W − 618. RT ( the distance ) = ( R − 8.5mi/h )(T + 2.4 h ) ⇒ R (T ) + R ( 2.4 ) − 8.5 (T ) − 8.5 ( 2.4 ) ⇒ RT + 2.4 R − 8.5T − 20.4 By calculator:Exercise 7• Product of Two Multimonials Multiply the following binomials and trinomials, and simplify. [See Examples 61 and 62] 1. ( x − 3)( x + 4 − y ) ⇒ x ( x ) + x ( 4 ) + x ( − y ) − 3 ( x ) − 3 ( 4 ) − 3 ( − y ) ⇒ x 2 + 4 x − xy − 3 x − 12 + 3 y= x 2 − xy + x + 3 y − 12Full file at https://fratstock.eu Chapter 22.( a − d )( a − 2d + 5 ) ⇒ a ( a ) + a ( −2d ) + a ( 5 ) − d ( a ) − d ( −2d ) − d ( 5 ) ⇒ a 2 − 2ad + 5a − ad + 2d 2 − 5d = a 2 − 3ad + 5a + 2d 2 − 5d3.(w2+ w − 5 ) ( 4 w − 2 ) ⇒ w2 ( 4w ) + w2 ( −2 ) + w ( 4 w ) + w ( −2 ) − 5 ( 4 w ) − 5 ( −2 )⇒ 4 w3 − 2 w2 + 4 w2 − 2 w − 20 w + 10 = 4 w3 − 2 w2 − 22 w + 104.(a2− 5 )( 3a 2 − 7 a − 4 ) ⇒ a 2 ( 3a 2 ) + a 2 ( −7 a ) + a 2 ( −4 ) − 5 ( 3a 2 ) − 5 ( −7 a ) − 5 ( −4 )⇒ 3a 4 − 7 a 3 − 4a 2 − 15a 2 + 35a + 20 = 3a 4 − 7 a 3 − 19a 2 + 35a + 205.(b7− 2.82b5 + 4.27b3 ) ( b + 2.93) ⇒ b 7 ( b ) + b 7 ( 2.93) − 2.82b5 ( b ) − 2.82b5 ( 2.93) + 4.27b3 ( b ) + 4.27b3 ( 2.93)⇒ b8 + 2.93b 7 − 2.82b 6 − 8.2626b 5 + 4.27b 4 + 12.5111b 3 = b8 + 2.93b 7 − 2.82b 6 − 8.26b 5 + 4.27b 4 + 12.5b 36.( x + 3.88) ( x3 − 2.15 x − 6.03) ⇒ x ( x3 ) + x ( −2.15 x ) + x ( −6.03) + 3.88 ( x3 ) + 3.88 ( −2.15 x ) + 3.88 ( −6.03) ⇒ x 4 − 2.15 x 2 − 6.03 x + 3.88 x 3 − 8.342 x − 23.3964 ⇒ x 4 + 3.88 x 3 − 2.15 x 2 − 14.372 x − 23.3964 = x 4 + 3.88 x 3 − 2.15 x 2 − 14.4 x − 23.47.(1 + c )( 4c 22+ 7c − 3) ⇒ 1( 4c 2 ) + 1( 7c ) − 1( 3) + c 2 ( 4c 2 ) + c 2 ( 7c ) − c 2 ( 3) ⇒ 4c 2 + 7c − 3 + 4c 4 + 7c3 − 3c 2= 4c 4 + 7 c 3 + c 2 + 7 c − 38.( 2x2− 6 xy + 3 y 2 ) ( 3x + 3 y ) ⇒ 2 x 2 ( 3x ) + 2 x 2 ( 3 y ) − 6 xy ( 3 x ) − 6 xy ( 3 y ) + 3 y 2 ( 3 x ) + 3 y 2 ( 3 y )⇒ 6 x3 + 6 x 2 y − 18 x 2 y − 18 xy 2 + 9 xy 2 + 9 y 3 = 6 x3 − 12 x 2 y − 9 xy 2 + 9 y 3 9.(b2− bx + x 2 ) ( b + x ) ⇒ b 2 ( b ) + b 2 ( x ) − bx ( b ) − bx ( x ) + x 2 ( b ) + x 2 ( x )⇒ b 3 + b 2 x − b 2 x − bx 2 + x 2b + x 3 = x 3 + b 310.(a2+ 2a − 2 ) ( a + 1) ⇒ a 2 ( a ) + a 2 (1) + 2a ( a ) + 2a (1) − 2 ( a ) − 2 (1) ⇒ a 3 + a 2 + 2a 2 + 2a − 2a − 2= a 3 + 3a 2 − 211.(b2+ bx + x 2 ) ( b − x ) ⇒ b 2 ( b ) + b 2 ( − x ) + bx ( b ) + bx ( − x ) + x 2 ( b ) + x 2 ( − x )⇒ b 3 − b 2 x + b 2 x − bx 2 + x 2b − x 3 = b3 − x 312.(b2+ bx − x 2 ) ( b − x ) ⇒ b 2 ( b ) + b 2 ( − x ) + bx ( b ) + bx ( − x ) − x 2 ( b ) − x 2 ( − x )⇒ b 3 − b 2 x + b 2 x − bx 2 − x 2b + x 3 = x 3 − 2bx 2 + b3Multiply the following binomials and polynomials and simplify [See Example 63] 13. ( a 2 + 5a − xy ) ( a + z ) ⇒ a 2 ( a ) + a 2 ( z ) + 5a ( a ) + 5a ( z ) − xy ( a ) − xy ( z ) ⇒ a 3 + a 2 z + 5a 2 + 5az − axy − xyz= a3 − xyz − axy + a 2 z + 5az + 5a 2 14.(c2− cm + cn + mn ) ( c − m ) ⇒ c 2 ( c ) + c 2 ( − m ) − cm ( c ) − cm ( − m ) + cn ( c ) + cn ( − m ) + mn ( c ) + mn ( − m )⇒ c 3 − c 2 m − c 2 m + cm 2 + c 2 n − cmn + cmn − m 2 n = c 3 − 2c 2 m + c 2 n + cm 2 − m 2 n15.(y2− x 2 )( y 3 + ay 2 − abxy + bx 2 − x3 )⇒ y 2 ( y 3 ) + y 2 ( ay 2 ) + y 2 ( − abxy ) + y 2 ( bx 2 ) + y 2 ( − x3 ) − x 2 ( y 3 ) − x 2 ( ay 2 ) − x 2 ( − abxy ) − x 2 ( bx 2 ) − x 2 ( − x3 )⇒ y 5 + ay 4 − abxy 3 + y 2bx 2 − x3 y 2 − x 2 y 3 − ax 2 y 2 + abx3 y − bx 4 + x545Full file at https://fratstock.eu 46Chapter 2= x5 − bx 4 − x3 y 2 + abx3 y − x 2 y 3 − ax 2 y 2 + bx 2 y 2 − abxy 3 + y 5 + ay 4 Multiply the following monomials and binomials and simplify [See Example 64] 16. 4 ( 5b + 3c )( 5b + 3c ) ⇒ 4 ⎡⎣5b ( 5b ) + 5b ( 3c ) + 3c ( 5b ) + 3c ( 3c ) ⎤⎦ ⇒ 4 ⎡⎣ 25b 2 + 15bc + 15bc + 9c 2 ⎤⎦ = 100b 2 + 120bc + 36c 217.( 4w − 5 z )( 4w − 5 z ) d ⇒ ⎡⎣ 4w ( 4w) + 4w ( −5 z ) − 5 z ( 4w) − 5 z ( −5 z )⎤⎦ d ⇒ ⎡⎣16w2 − 20wz − 20wz + 25 z 2 ⎤⎦ d = 16dw2 − 40 dwz + 25dz 218. z ( 22.1a − 3.03b )( 2.26a + 38.2b ) ⇒ z ⎡⎣ 22.1a ( 2.26a ) + 22.1a ( 38.2b ) − 3.03b ( 2.26a ) − 3.03b ( 38.2b )⎤⎦ ⇒ z ⎣⎡ 49.946a 2 + 844.22ab − 6.8478ab − 115.746b 2 ⎦⎤ ⇒ z ⎡⎣ 49.946a 2 + 837.37 ab − 115.746b 2 ⎤⎦ = 49.946a 2 z + 837.37 abz − 115.746b 2 z19.( d − 4.11)( d − 4.93)( d + 2.26 ) ⇒ ⎡⎣d ( d ) + d ( −4.93) − 4.11( d ) − 4.11( −4.93)⎤⎦ ⎡⎣( d + 2.26 )⎤⎦ ⇒ d ( d )( d ) + d ( −4.93)( d ) − 4.11( d )( d ) − 4.11( −4.93)( d
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