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3.4 Übungen

Aus Online Mathematik Brückenkurs 2

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{{Ej vald flik|[[3.4 Komplexa polynom|Teori]]}}
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{{Ej vald flik|[[3.4 Komplexa polynom|Theory]]}}
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{{Vald flik|[[3.4 Övningar|Övningar]]}}
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{{Vald flik|[[3.4 Övningar|Exercises]]}}
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===Övning 3.4:1===
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===Exercise 3.4:1===
<div class="ovning">
<div class="ovning">
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Utför följande polynomdivisioner (alla går inte jämnt ut)
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Carry out the following division:(not all are exact, i.e. have no remainder)
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
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|a)
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</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}}
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===Övning 3.4:2===
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===Exercise 3.4:2===
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<div class="ovning">
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Ekvationen <math>\,z^3-3z^2+4z-2=0\,</math> har roten <math>\,z=1\,</math>. Bestäm övriga rötter.
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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}}
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===Övning 3.4:3===
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===Exercise 3.4:3===
<div class="ovning">
<div class="ovning">
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Ekvationen <math>\,z^4+2z^3+6z^2 +8z +8 =0\,</math> har rötterna <math>\,z=2i\,</math> och <math>\,z=-1-i\,</math>. Lös ekvationen.
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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}}
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===Övning 3.4:4===
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===Exercise 3.4:4===
<div class="ovning">
<div class="ovning">
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Bestäm två reella tal <math>\,a\,</math> och <math>\,b\,</math> så att ekvationen <math>\ z^3+az+b=0\ </math> har roten <math>\,z=1-2i\,</math>. Lös sedan ekvationen.
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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}}
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===Övning 3.4:5===
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===Exercise 3.4:5===
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Bestäm <math>\,a\,</math> och <math>\,b\,</math> så att ekvationen <math>\ z^4-6z^2+az+b=0\ </math> har en trippelrot. Lös sedan ekvationen.
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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}}
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===Övning 3.4:6===
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===Exercise 3.4:6===
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<div class="ovning">
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Ekvationen <math>\ z^4+3z^3+z^2+18z-30=0\ </math> har en rent imaginär rot. Bestäm alla rötter.
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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}}
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===Övning 3.4:7===
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===Exercise 3.4:7===
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<div class="ovning">
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Bestäm polynom som har följande nollställen
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Determine the polynomial which has the following zeros
{| width="100%" cellspacing="10px"
{| width="100%" cellspacing="10px"
|a)
|a)
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|width="50%"|<math>1\,</math>, <math>\,2\,</math> och <math>\,4</math>
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|width="50%"|<math>1\,</math>, <math>\,2\,</math> and <math>\,4</math>
|b)
|b)
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|width="50%"| <math>-1+ i\,</math> och <math>\,-1-i</math>
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|width="50%"| <math>-1+ i\,</math> and <math>\,-1-i</math>
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|}
</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) x−1x2−1 b) x2x+1 c) x+ax3+a3
d) x+1x3+x+2 e) x2+3x+1x3+2x2+1

Svar | Lösning a | Lösning b | Lösning c | Lösning d | Lösning e

Exercise 3.4:2

The equation z3−3z2+4z−2=0 has the root z=1. Determine the other roots.

Svar | Lösning

Exercise 3.4:3

The equation z4+2z3+6z2+8z+8=0 has the roots z=2i and z=−1−i. Solve the equation.


Svar | Lösning

Exercise 3.4:4

Determine two real numbers a and b , such that the equation  z3+az+b=0  has the root z=1−2i. Then solve the equation.


Svar | Lösning

Exercise 3.4:5

Determine a and 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

Svar | Lösning a | Lösning b

Von „http://wiki.math.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/3.4_%C3%9Cbungen“