Design and use of a learning object for finding complex polynomial roots

Julio Benítez, José L. Hueso, Eulalia Martínez, Jaime Riera

Roots of the same modulus

The three roots of the polynomial $p(z)= (z-1)(z-i)(z+i)=z^3+z^2+z+1$ have the same modulus, 1.

Define this polynomial in the applet by entering its coefficients in the Algebra panel or by dragging the corresponding points in the Graphics panel to:

  • $a_0 = 1+0i$
  • $a_1 = 1+0i$
  • $a_2 = 1+0i$
  • $a_3 = 1+0i$
  • $a_n = 0+0i$

Set now $r$ to a small value. Observe in the second window that the image curve does not include the origin in its interior.


Increase gradually the value of $r$ using the corresponding slider in the image window. The three branches of the curve simultaneously cross the origin, when $r=1.$


Now, the three roots of p can be found only dragging the $\theta$ slider.

Exercise

Find the roots of the polynomial $z^5-1.$

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