3.4 Exercises

From Förberedande kurs i matematik 2

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{{Mall:Ej vald flik|[[3.4 Komplexa polynom|Teori]]}}
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===Övning 3.4:1===
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===Exercise 3.4:1===
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Utför följande polynomdivisioner (alla går inte jämnt ut)
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Carry out the following divisions (not all are exact, i.e. have no remainder)
{| width="100%" cellspacing="10px"
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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: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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</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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===Övning 3.4:2===
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===Exercise 3.4:2===
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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.
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</div>{{#NAVCONTENT:Svar|Svar 3.4:2|Lösning |Lösning 3.4:2}}
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</div>{{#NAVCONTENT:Answer|Answer 3.4:2|Solution|Solution 3.4:2}}
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===Övning 3.4:3===
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===Exercise 3.4:3===
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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}}
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</div>{{#NAVCONTENT:Answer|Answer 3.4:3|Solution|Solution 3.4:3}}
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===Övning 3.4:4===
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===Exercise 3.4:4===
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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}}
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</div>{{#NAVCONTENT:Answer|Answer 3.4:4|Solution|Solution 3.4:4}}
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===Exercise 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.
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</div>{{#NAVCONTENT:Answer|Answer 3.4:5|Solution|Solution 3.4:5}}
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===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.
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</div>{{#NAVCONTENT:Answer|Answer 3.4:6|Solution|Solution 3.4:6}}
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===Exercise 3.4:7===
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Determine the polynomial which has the following zeros
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|a)
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|width="50%"|<math>1\,</math>, <math>\,2\,</math> and <math>\,4</math>
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|b)
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|width="50%"| <math>-1+ i\,</math> and <math>\,-1-i</math>
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|}
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</div>{{#NAVCONTENT:Answer|Answer 3.4:7|Solution a|Solution 3.4:7a|Solution b|Solution 3.4:7b}}

Current revision

       Theory          Exercises      

Exercise 3.4:1

Carry out the following divisions (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}
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

The equation \displaystyle \,z^3-3z^2+4z-2=0\, has the root \displaystyle \,z=1\,. Determine the other roots.

Answer
Answer 3.4:2
Solution
Solution 3.4:2

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.

Answer
Answer 3.4:3
Solution
Solution 3.4:3

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.

Answer
Answer 3.4:4
Solution
Solution 3.4:4

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.

Answer
Answer 3.4:5
Solution
Solution 3.4:5

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
Answer 3.4:6
Solution
Solution 3.4:6

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
Answer 3.4:7
Solution a
Solution 3.4:7a
Solution b
Solution 3.4:7b
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