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3.4 Übungen

Aus Online Mathematik Brückenkurs 2

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K (Robot: Automated text replacement (-Svar +Answer))
K (Robot: Automated text replacement (-Lösning +Solution))
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|width="33%"| <math>\displaystyle \frac{x^3+2x^2+1}{x^2+3x+1}</math>
|width="33%"| <math>\displaystyle \frac{x^3+2x^2+1}{x^2+3x+1}</math>
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</div>{{#NAVCONTENT:Answer|Answer 3.4:1|Solution a|Lösning 3.4:1a|Solution b|Lösning 3.4:1b|Solution c|Lösning 3.4:1c|Solution d|Lösning 3.4:1d|Solution e|Lösning 3.4:1e}}
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</div>{{#NAVCONTENT:Answer|Answer 3.4:1|Solution a|Solution 3.4:1a|Solution b|Solution 3.4:1b|Solution c|Solution 3.4:1c|Solution d|Solution 3.4:1d|Solution e|Solution 3.4:1e}}
===Exercise 3.4:2===
===Exercise 3.4:2===
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<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.
The equation <math>\,z^3-3z^2+4z-2=0\,</math> has the root <math>\,z=1\,</math>. Determine the other roots.
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</div>{{#NAVCONTENT:Answer|Answer 3.4:2|Solution|Lösning 3.4:2}}
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</div>{{#NAVCONTENT:Answer|Answer 3.4:2|Solution|Solution 3.4:2}}
===Exercise 3.4:3===
===Exercise 3.4:3===
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<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.
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:Answer|Answer 3.4:3|Solution|Lösning 3.4:3}}
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</div>{{#NAVCONTENT:Answer|Answer 3.4:3|Solution|Solution 3.4:3}}
===Exercise 3.4:4===
===Exercise 3.4:4===
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<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.
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:Answer|Answer 3.4:4|Solution|Lösning 3.4:4}}
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</div>{{#NAVCONTENT:Answer|Answer 3.4:4|Solution|Solution 3.4:4}}
===Exercise 3.4:5===
===Exercise 3.4:5===
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<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.
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.
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</div>{{#NAVCONTENT:Answer|Answer 3.4:5|Solution|Lösning 3.4:5}}
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</div>{{#NAVCONTENT:Answer|Answer 3.4:5|Solution|Solution 3.4:5}}
===Exercise 3.4:6===
===Exercise 3.4:6===
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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.
The equation <math>\ z^4+3z^3+z^2+18z-30=0\ </math> has a pure imaginary root. Determine all the roots.
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</div>{{#NAVCONTENT:Answer|Answer 3.4:6|Solution|Lösning 3.4:6}}
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</div>{{#NAVCONTENT:Answer|Answer 3.4:6|Solution|Solution 3.4:6}}
===Exercise 3.4:7===
===Exercise 3.4:7===
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|width="50%"| <math>-1+ i\,</math> and <math>\,-1-i</math>
|width="50%"| <math>-1+ i\,</math> and <math>\,-1-i</math>
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</div>{{#NAVCONTENT:Answer|Answer 3.4:7|Solution a|Lösning 3.4:7a|Solution b|Lösning 3.4:7b}}
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</div>{{#NAVCONTENT:Answer|Answer 3.4:7|Solution a|Solution 3.4:7a|Solution b|Solution 3.4:7b}}

Version vom 07:31, 17. Sep. 2008

 
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Exercise 3.4:1

Carry out the following divisions (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

Answer | Solution a | Solution b | Solution c | Solution d | Solution e

Exercise 3.4:2

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

Answer | Solution

Exercise 3.4:3

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

Answer | Solution

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.

Answer | Solution

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.

Answer | Solution

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.

Answer | Solution

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

Answer | Solution a | Solution b

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