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Complex Numbers and Polynomials - Questions

Question 1
26293

Given that \(2+i\) is a zero of \(P(z)=z^3-3z^2+z+5\), find the other zeros of \(P(2)\).

zeros are \(-1,2+i,2-i\)

Question 2
21639

Factorise \({z^4} + {z^2} - 12\) over the complex field. 

\((z + 2i)(z - 2i)(z + \sqrt 3 )(z - \sqrt 3 )\)

Question 3
17250

Given that \({z^n} + {z^{ - n}} = 2\cos n\theta \), then the solutions to \(3{z^4} + {z^3} + 4{z^2} + z + 3 = 0\) are?

\(z = \dfrac{{-1 \pm i\sqrt 3 }}{2},\dfrac{{1 \pm i\sqrt 8 }}{3}\)

Question 4
17251

The factors of \({z^3} + {z^2} + z + 1\) are?

\(\left( {z + 1} \right)\left( {z - i} \right)\left( {z + i} \right)\)

Question 5
12836

Given that \(\sin 3\theta = 3\sin \theta - 4{\sin ^3}\theta ,{\rm{ }}\) then the solutions to \(8{x^3} - 6x + 1 = 0 \) are?

\(x = \sin \dfrac{\pi }{{18}},\,\sin \dfrac{{5\pi }}{{18}},\,\, - \sin \dfrac{{7\pi }}{{18}}\)

Question 6
12837

Given that \({\rm{ }}{z^n} + {z^{ - n}} = 2\cos n\theta ,{\rm{ }}\) then the solutions to \({\rm{ 2}}{{\rm{z}}^4} + 3{z^3} + 5{z^2} + 3z + 2 = 0\) are ?

\(z = \dfrac{{ - 1 \pm i\sqrt 3 }}{2},\dfrac{{ - 1 \pm i\sqrt {15} }}{4}\)

Question 7
12263

The factors of \({z^2} + z + 1\) are?

\(\left( {z + \dfrac{{1 - i\sqrt 3 }}{2}} \right)\left( {z + \dfrac{{1 + i\sqrt 3 }}{2}} \right)\)

Question 8
12264

The zeros of \(2{z^3} - {z^2} + 18z - 9 = 0\) are?

\(z = \dfrac{1}{2}, \pm \,3i\)

Question 9
12265

Given that \(z = 2 - 3i\) is a zero of \(P(z) = {z^3} + {z^2} - 7z + 65\) then the other two zeros are?

\({z_2} = 2 + 3i,\,{z_3} = - 5\)

Question 10
12266

The zeros of \({z^6} - 1 = 0\,\) are?

\( \pm 1,\,\dfrac{{ - 1 \pm i\sqrt 3 }}{2},\,\,\dfrac{{1 \pm i\sqrt 3 }}{2}\)

Question 11
12267

The factors of \({z^4} + {z^2} + 1\) are?

\(\left( {z + \left[ {\dfrac{{1 - i\sqrt 3 }}{2}} \right]} \right)\left( {z + \left[ {\dfrac{{1 + i\sqrt 3 }}{2}} \right]} \right)\left( {z - \left[ {\dfrac{{1 + i\sqrt 3 }}{2}} \right]} \right)\left( {z - \left[ {\dfrac{{1 - i\sqrt 3 }}{2}} \right]} \right)\)

Question 12
52079

Find all the zeros of the polynomial \(P(z)=z^4-z^3+6z^2-z+15\), given that \(1-2i\) is a zero of \(P(z)\)