Introduction Notes Theory HOWTO Examples Engineering Error Questions Matlab Maple

Example 1

In Matlab, divide out the root 0.55169 + 1.25335j and its complex conjugate from the polynomial x⁵ + 2x⁴ + 3x³ + 4x² + 5x + 6 without involving complex numbers.

The quadratic which has the two given roots is (x - (0.55169 + 1.25335j))(x - (0.55169 - 1.25335j)) = x² - 2 0.55169 x + 0.55169² + 1.25335² = x² - 1.10338x + 1.87525.

This is represented by the vector [1 -1.10338 1.87525], and thus we calculate:

>> roots( [1 2 3 4 5 6] )                       
ans =

   0.55169 + 1.25335i
   0.55169 - 1.25335i
  -0.80579 + 1.22290i
  -0.80579 - 1.22290i
  -1.49180 + 0.00000i

>> deconv( [1 2 3 4 5 6], [1 -1.10338 1.87525] )
ans =

  1.00000  3.10338  4.54896  3.19962

>> roots( ans )                                 
ans =

  -1.49180 + 0.00000i
  -0.80579 + 1.22291i
  -0.80579 - 1.22291i

The remaining roots are coloured to highlight that they remained unchanged throughout the division.

Example 2

What happens if you try to divide out values which are not close to the actual roots? From Example 1, divide out the false root 1 + j and its complex conjugate, i.e., (x − (1 + j))(x − (1 − j)) = x² − 2x + 2.

If we divide out the two given roots, as follows:

>> roots( [1 2 3 4 5 6] ) 
ans =
   0.55169 + 1.25335i
   0.55169 - 1.25335i
  -0.80579 + 1.22290i
  -0.80579 - 1.22290i
  -1.49180 + 0.00000i

>> deconv( [1 2 3 4 5 6], [1 -2 2] )
ans =
        1        4        9       14

>> roots( ans )                     
ans =
  -2.60752 + 0.00000i
  -0.69624 + 2.21005i
  -0.69624 - 2.21005i

we note that Matlab will perform the division, but the roots of the solution are significantly different from the roots of the original polynomial.

Topic 10.5: Polynomials (Examples)

Example 1

Example 2