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3.4 Übungen

Aus Online Mathematik Brückenkurs 2

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===Exercise 3.4:1===
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===Übung 3.4:1===
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Carry out the following divisions (not all are exact, i.e. have no remainder)
Carry out the following divisions (not all are exact, i.e. have no remainder)
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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}}
</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}}
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===Exercise 3.4:2===
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===Übung 3.4:2===
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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.
The equation <math>\,z^3-3z^2+4z-2=0\,</math> has the root <math>\,z=1\,</math>. Determine the other roots.
</div>{{#NAVCONTENT:Answer|Answer 3.4:2|Solution|Solution 3.4:2}}
</div>{{#NAVCONTENT:Answer|Answer 3.4:2|Solution|Solution 3.4:2}}
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===Exercise 3.4:3===
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===Übung 3.4:3===
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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.
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.
</div>{{#NAVCONTENT:Answer|Answer 3.4:3|Solution|Solution 3.4:3}}
</div>{{#NAVCONTENT:Answer|Answer 3.4:3|Solution|Solution 3.4:3}}
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===Exercise 3.4:4===
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===Übung 3.4:4===
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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.
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.
</div>{{#NAVCONTENT:Answer|Answer 3.4:4|Solution|Solution 3.4:4}}
</div>{{#NAVCONTENT:Answer|Answer 3.4:4|Solution|Solution 3.4:4}}
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===Exercise 3.4:5===
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===Übung 3.4:5===
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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.
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:Answer|Answer 3.4:5|Solution|Solution 3.4:5}}
</div>{{#NAVCONTENT:Answer|Answer 3.4:5|Solution|Solution 3.4:5}}
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===Exercise 3.4:6===
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===Übung 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.
</div>{{#NAVCONTENT:Answer|Answer 3.4:6|Solution|Solution 3.4:6}}
</div>{{#NAVCONTENT:Answer|Answer 3.4:6|Solution|Solution 3.4:6}}
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===Exercise 3.4:7===
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===Übung 3.4:7===
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Determine the polynomial which has the following zeros
Determine the polynomial which has the following zeros

Version vom 13:24, 10. Mär. 2009

       Theory          Übungen      

Übung 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

Übung 3.4:2

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

Answer | Solution

Übung 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

Übung 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

Übung 3.4:5

Determine a and b so that the equation  z4−6z2+az+b=0  has a triple root. Then solve the equation.

Answer | Solution

Übung 3.4:6

The equation  z4+3z3+z2+18z−30=0  has a pure imaginary root. Determine all the roots.

Answer | Solution

Übung 3.4:7

Determine the polynomial which has the following zeros

a) 1, 2 and 4 b) −1+i and −1−i

Answer | Solution a | Solution b

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