3.4 Übungen
Aus Online Mathematik Brückenkurs 2
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| - | {{Ej vald flik|[[3.4 Komplexa polynom| | + | {{Ej vald flik|[[3.4 Komplexa polynom|Theory]]}} |
| - | {{Vald flik|[[3.4 Övningar| | + | {{Vald flik|[[3.4 Övningar|Exercises]]}} |
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| - | === | + | ===Exercise 3.4:1=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | Carry out the following division:(not all are exact, i.e. have no remainder) | |
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
|a) | |a) | ||
| Zeile 25: | Zeile 25: | ||
</div>{{#NAVCONTENT:Svar|Svar 3.4:1|Lösning a|Lösning 3.4:1a|Lösning b|Lösning 3.4:1b|Lösning c|Lösning 3.4:1c|Lösning d|Lösning 3.4:1d|Lösning e|Lösning 3.4:1e}} | </div>{{#NAVCONTENT:Svar|Svar 3.4:1|Lösning a|Lösning 3.4:1a|Lösning b|Lösning 3.4:1b|Lösning c|Lösning 3.4:1c|Lösning d|Lösning 3.4:1d|Lösning e|Lösning 3.4:1e}} | ||
| - | === | + | ===Exercise 3.4:2=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | The equation <math>\,z^3-3z^2+4z-2=0\,</math> has the root <math>\,z=1\,</math>. Determine the other roots. | |
</div>{{#NAVCONTENT:Svar|Svar 3.4:2|Lösning |Lösning 3.4:2}} | </div>{{#NAVCONTENT:Svar|Svar 3.4:2|Lösning |Lösning 3.4:2}} | ||
| - | === | + | ===Exercise 3.4:3=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | The equation <math>\,z^4+2z^3+6z^2 +8z +8 =0\,</math> has the roots <math>\,z=2i\,</math> and <math>\,z=-1-i\,</math>. Solve the equation. | |
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</div>{{#NAVCONTENT:Svar|Svar 3.4:3|Lösning |Lösning 3.4:3}} | </div>{{#NAVCONTENT:Svar|Svar 3.4:3|Lösning |Lösning 3.4:3}} | ||
| - | === | + | ===Exercise 3.4:4=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | Determine two real numbers <math>\,a\,</math> and <math>\,b\,</math> , such that the equation <math>\ z^3+az+b=0\ </math> has the root <math>\,z=1-2i\,</math>. Then solve the equation. | |
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</div>{{#NAVCONTENT:Svar|Svar 3.4:4|Lösning |Lösning 3.4:4}} | </div>{{#NAVCONTENT:Svar|Svar 3.4:4|Lösning |Lösning 3.4:4}} | ||
| - | === | + | ===Exercise 3.4:5=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | Determine <math>\,a\,</math> and <math>\,b\,</math> so that the equation <math>\ z^4-6z^2+az+b=0\ </math> has a triple root. Then solve the equation. | |
</div>{{#NAVCONTENT:Svar|Svar 3.4:5|Lösning |Lösning 3.4:5}} | </div>{{#NAVCONTENT:Svar|Svar 3.4:5|Lösning |Lösning 3.4:5}} | ||
| - | === | + | ===Exercise 3.4:6=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | The equation <math>\ z^4+3z^3+z^2+18z-30=0\ </math> has a pure imaginary root. Determine all the roots. | |
</div>{{#NAVCONTENT:Svar|Svar 3.4:6|Lösning |Lösning 3.4:6}} | </div>{{#NAVCONTENT:Svar|Svar 3.4:6|Lösning |Lösning 3.4:6}} | ||
| - | === | + | ===Exercise 3.4:7=== |
<div class="ovning"> | <div class="ovning"> | ||
| - | + | Determine the polynomial which has the following zeros | |
{| width="100%" cellspacing="10px" | {| width="100%" cellspacing="10px" | ||
|a) | |a) | ||
| - | |width="50%"|<math>1\,</math>, <math>\,2\,</math> | + | |width="50%"|<math>1\,</math>, <math>\,2\,</math> and <math>\,4</math> |
|b) | |b) | ||
| - | |width="50%"| <math>-1+ i\,</math> | + | |width="50%"| <math>-1+ i\,</math> and <math>\,-1-i</math> |
|} | |} | ||
</div>{{#NAVCONTENT:Svar|Svar 3.4:7|Lösning a|Lösning 3.4:7a|Lösning b|Lösning 3.4:7b}} | </div>{{#NAVCONTENT:Svar|Svar 3.4:7|Lösning a|Lösning 3.4:7a|Lösning b|Lösning 3.4:7b}} | ||
Version vom 12:46, 4. Aug. 2008
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Exercise 3.4:1
Carry out the following division:(not all are exact, i.e. have no remainder)
| a) | \displaystyle \displaystyle\frac{x^2-1}{x-1} | b) | \displaystyle \displaystyle\frac{x^2}{x+1} | c) | \displaystyle \displaystyle \frac{x^3+a^3}{x+a} |
| d) | \displaystyle \displaystyle\frac{x^3 +x+2}{x+1} | e) | \displaystyle \displaystyle \frac{x^3+2x^2+1}{x^2+3x+1} |
Svar
Laden...
Lösning a
Lösning b
Lösning c
Lösning d
Lösning e
Exercise 3.4:2
The equation \displaystyle \,z^3-3z^2+4z-2=0\, has the root \displaystyle \,z=1\,. Determine the other roots.
Svar
Lösning
Exercise 3.4:3
The equation \displaystyle \,z^4+2z^3+6z^2 +8z +8 =0\, has the roots \displaystyle \,z=2i\, and \displaystyle \,z=-1-i\,. Solve the equation.
Svar
Lösning
Exercise 3.4:4
Determine two real numbers \displaystyle \,a\, and \displaystyle \,b\, , such that the equation \displaystyle \ z^3+az+b=0\ has the root \displaystyle \,z=1-2i\,. Then solve the equation.
Svar
Lösning
Exercise 3.4:5
Determine \displaystyle \,a\, and \displaystyle \,b\, so that the equation \displaystyle \ z^4-6z^2+az+b=0\ has a triple root. Then solve the equation.
Svar
Lösning
Exercise 3.4:6
The equation \displaystyle \ z^4+3z^3+z^2+18z-30=0\ has a pure imaginary root. Determine all the roots.
Svar
Lösning
Exercise 3.4:7
Determine the polynomial which has the following zeros
| a) | \displaystyle 1\,, \displaystyle \,2\, and \displaystyle \,4 | b) | \displaystyle -1+ i\, and \displaystyle \,-1-i |
