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| 1. | a. | 8x3
+ 36x2 + 54x + 19 = 0 met x = y + c
geeft: 8(y + c)3 + 36(y + c)2 + 54(y + c) + 19 = 0 8(y3 + 3y2c + 3yc2 + c3) + 36(y2 + 2yc + c2 ) + 54y + 54c + 19 = 0 8y3 + 24y2c + 24yc2 + 8c3 + 36y2 + 72yc + 36c2 + 54y + 54c + 19 = 0 8y3 + y2(24c + 36) + y(24c2 + 72c + 54) + (8c3 + 36c2 + 54c + 19) = 0 Kies nu 24c + 36 = 0 dus c = - 1,5 Dat geeft y3 - 8 = 0 y3 = 8 y = 2 x = 2 - 1,5 = 0,5 |
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| b. | 1 - 12x + 48x2 - 91x3
= 0 met x = 1/(y
+ 4) geeft; 1 - 12/(y + 4) + 48/(y + 4)2 - 91/(y + 4)3 = 0 vermenigvuldig alles met (y + 4)3: (y + 4)3 - 12(y + 4)2 + 48(y + 4) - 91 = 0 y3 + 12y2 + 48y + 64 - 12y2 - 96y - 192 + 48y + 192 - 91 = 0 y3 - 27 = 0 y3 = 27 y = 3 en dan is x = 1/(4 - 3) = 1 |
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| c. | x2
+ 16xÖx + 96x +
256Öx - 1040 = 0 met x = (y - 4)2
geeft: (y - 4)4 + 16(y - 4)3 + 96(y - 4)2 + 256(y - 4) - 1040 = 0 y4 - 16y3 + 96y2 - 256y + 256 + 16y3 - 192y2 + 768y - 1024 + 96y2 - 768y + 1536+256y-1024 -1040=0 y4 - 1296 = 0 y4 = 1296 y = 6 x = (y - 4)2 = 4 |
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| d. | x3 -
11x2 + 35x - 25 = 0 met y = x - p
geeft: (y + p)3 - 11(y + p)2 + 35(y + p) - 25 = 0 y3 + 3y2p + 3yp2 + p3 - 11y2 - 22yp - 11p2 + 35y + 35p - 25 = 0 y3 + y2(3p - 11) + y(3p2 - 22p + 35) + (p3 - 11p2 + 35p - 25) = 0 p = 5 laat de laatste twee temen wegvallen: y3 + 4y2 = 0 y2(y + 4) = 0 y = 0 ∨ y = -4 Dan is x = 5 ∨ x = 1 |
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| 2. | a. | z3 + 3z2
+ 9z + 14 = 0 z = y - 1 geeft: (y - 1)3 + 3(y - 1)2 + 9(y - 1) + 14 = 0 y3 - 3y2 + 3y - 1 + 3y2 - 6y + 3 + 9y - 9 + 14 = 0 y3 + 6y + 7 = 0 y = m + n geeft: (m + n)3 + 6(m + n) + 7 = 0 m3 + 3m2n + 3mn2 + n3 + 6m + 6n + 7 = 0 m3 + n3 + 3mn(m + n) + 6(m + n) + 7 = 0 m3 + n3 + 7 + (m + n)(3mn + 6) = 0 kies nu 3mn = -6 ofwel n = -2/m Dat geeft m3 - 8/m3 + 7 = 0 m6 - 8 + 7m3 = 0 noem nu m3 = p: p2 + 7p - 8 = 0 geeft p = 1 ∨ p = -8 Dan is m3 = 1 m3 = -8 ofwel m = 1 ∨ m = -2 Dan is n = -2 ∨ n = 1 Beiden geeft y = 1 + -2 = -1 Dan is z = -2 |
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| b. | z3 + 3z2
+ 3z + 2 = 0 z = y - 1 geeft: (y - 1)3 + 3(y - 1)2 + 3(y - 1) + 2 = 0 y3 - 3y2 + 3y - 1 + 3y2 - 6y + 3 + 3y - 3 + 2 = 0 y3 + 1 = 0 Mazzel!! y3 = -1 Dat geeft direct y = -1 en dan is z = -2 |
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| c. | z3 + 3z2
-
9z -
27 = 0 z = y - 1 geeft: (y - 1)3 + 3(y - 1)2 - 9(y - 1) - 27 = 0 y3 - 3y2 + 3y - 1 + 3y2 - 6y + 3 - 9y + 9 - 27 = 0 y3 - 12y - 16 = 0 y = m + n geeft: (m + n)3 - 12(m + n) - 16 = 0 m3 + 3m2n + 3mn2 + n3 - 12m - 12n - 16 = 0 m3 + n3 - 16 + 3mn(m + n) - 12(m + n) = 0 m3 + n3 - 16 + (3mn - 12)(m + n) = 0 kies nu 3mn = 12 dus n = 4/m m3 + 64/m3 - 16 = 0 m6 + 64 - 16m3 = 0 m6 - 16m3 + 64 = 0 Noem nu m3 = p, dat geeft: p2 - 16p + 64 = 0 (p - 8)2 = 0 p = 8 Dan is m = 2 en n = 2 Dan is y = 4 Dan is z = 3 |
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| d. | 64z3 - 48z2
+ 12z - 1 = 0 z3 - 3/4z2 + 3/16z - 1/64 = 0 z = y + 1/4 geeft: (y + 1/4)3 - 3/4(y + 1/4)2 + 3/16(y + 1/4) - 1/64 = 0 y3 + 3/4y2 + 3/16y + 1/64 - 3/4y2 - 3/8y - 3/64 + 3/16y + 3/64 - 1/64 = 0 y3 = 0 y = 0 z = 1/4 |
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| e. | z3 + 6z2
+ 11z + 6 = 0 z = y - 2 geeft: (y - 2)3 + 6(y - 2)2 + 11(y - 2) + 6 = 0 y3 - 6y2 + 12y - 8 + 6y2 - 24y + 24 + 11y - 22 + 6 = 0 y3 - y = 0 y(y2 - 1) = 0 y = 0 ∨ y = 1 ∨ y = -1 Dat geeft z = -2 ∨ z = -1 ∨ z = -3 |
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| f. | z3
+ 6z2 + 12z + 9 = 0 z = y - 2 geeft: (y - 2)3 + 6(y - 2)2 + 12(y - 2) + 9 = 0 y3 - 6y2 + 12y - 8 + 6y2 - 24y + 24 + 12y - 24 + 9 = 0 y3 + 1 = 0 y3 = -1 y = -1 Dan is z = -3 |
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| 3. | a. | z3 + 2z2
- 14z - 40 = 0 z = y - 2/3 geeft: (y - 2/3)3 + 2(y - 2/3)2 - 14(y - 2/3) - 40 = 0 y3 - 2y2 + 12/9y - 8/27 + 2y2 - 8/3y + 8/9 - 14y + 28/3 - 40 = 0 y3 - 46/3y - 812/27 = 0 y = m + n geeft: (m + n)3 - 46/3(m + n) - 812/27 = 0 m3 + 3m2n + 3mn2 + n3 - 46/3m - 46/3n - 812/27 = 0 m3 + n3 - 812/27 + 3mn(m + n) - 46/3(m + n) = 0 m3 + n3 - 812/27 + (m + n)(3mn + 46/3) = 0 kies nu 3mn = 46/3 dus n = 46/9m m3 + (97336/729m3) - 812/27 = 0 729m6 + 97336 - 21924m3 = 0 D = 219242 - 4 • 729 • 97336 = 196830000 m3 , n3 = (21924 ± √(196830000))/1458 Dan is y = m + n = 42/3 Dan is z = 4 |
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| b. | z3
+ 3z2 + 4z - 28 = 0 z = y - 1 geeft: (y - 1)3 + 3(y - 1)2 + 4(y - 1) - 28 = 0 y3 - 3y2 + 3y - 1 + 3y2 - 6y + 3 + 4y - 4 - 28 = 0 y3 + y - 30 = 0 y = m + n geeft (m + n)3 + (m + n) - 30 = 0 m3 + 3m2n + 3mn2 + n3 + m + n - 30 = 0 m3 + n3 - 30 + 3mn(m + n) + m + n = 0 m3 + n3 - 30 + (3mn + 1)(m + n) = 0 kies nu 3mn + 1 = 0 dus n = -1/3m m3 + -1/27m3 - 30 = 0 27m6 - 1 - 810m3 = 0 D = 8102 + 4 • 27 = 656208 m3, n3 = (810 ± √656208)/54 Dan is y = m + n = 3 Dan is z = 2 |
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© h.hofstede (h.hofstede@hogeland.nl) |
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