How do you find all the real and complex roots of #f (x) = x^5 – x^4 + 3x^3 + 9x^2 – x + 5#?

1 Answer
May 31, 2016

Answer:

Use the Durand-Kerner method to find numerical approximations for the zeros.

Explanation:

Given:

#f(x) = x^5-x^4+3x^3+9x^2-x+5#

By the rational root theorem, any rational zeros of #f(x)# are expressible in the form #p/q# for integers #p, q# with #p# a divisor of the constant term #5# and #q# a divisor of the coefficient #1# of the leading term.

So the only possible rational zeros are:

#+-1#, #+-5#

None of these work, so the zeros of #f(x)# are irrational and/or Complex.

To find numeric approximations for the zeros, you can use the Durand-Kerner method:

Suppose the #5# zeros are: #p, q, r, s, t#.

Choose initial approximations for these zeros as follows:

#p_0 = (0.4+0.9i)^0#

#q_0 = (0.4+0.9i)^1#

#r_0 = (0.4+0.9i)^2#

#s_0 = (0.4+0.9i)^3#

#t_0 = (0.4+0.9i)^4#

Then iterate using the formulas:

#p_(i+1) = p_i-(f(p_i))/((p_i-q_i)(p_i-r_i)(p_i-s_i)(p_i-t_i))#

#q_(i+1) = q_i-(f(q_i))/((q_i-p_(i+1))(q_i-r_i)(q_i-s_i)(q_i-t_i))#

#r_(i+1) = r_i-(f(r_i))/((r_i-p_(i+1))(r_i-q_(i+1))(r_i-s_i)(r_i-t_i))#

#s_(i+1) = s_i-(f(s_i))/((s_i-p_(i+1))(s_i-q_(i+1))(s_i-r_(i+1))(s_i-t_i))#

#t_(i+1) = t_i-(f(t_i))/((t_i-p_(i+1))(t_i-q_(i+1))(t_i-r_(i+1))(t_i-s_(i+1))#

Keep iterating until the values are stable to the desired accuracy.

The zeros you will find will be approximately:

#-1.6073#

#0.109713+0.702004i#

#0.109713-0.702004i#

#1.19394+2.17633i#

#1.19394-2.17633i#

Here's a C++ program which does the calculation:

enter image source here

This is what its (slightly primitive) output looks like:

enter image source here