Question

How do you find the zeroes for `f(x)=x^5+3x^3-x+6`?

Answer 1

Use a graphing calculator.

Explanation:

Use a graphing calculator.

graph{x^5+3x^3-x+6 [-15.09, 16.95, -5.13, 10.89]}

There's only one zero: `(-1.178,0)`

Answer 2

Use Newton's method to find Real root:

`x ~~ -1.178259848`

and Complex roots:

`x ~~ -0.20255440359096+-1.893126314652i`

`x ~~ 0.79168432752598+-0.882050595027i`

Explanation:

`f(x) = x^5+3x^3-x+6`

`f'(x) = 5x^4+9x^2-1`

Starting with a first approximation `a_0`, iterate using the formula:

`a_(i+1) = a_i - (f(a_i))/(f'(a_i))`

Putting this into a spreadsheet, choosing `a_0 = -1`, we find:

`a_0 = -1`

`a_1 = -1.230769231`

`a_2 = -1.181552182`

`a_3 = -1.178273619`

`a_4 = -1.178259848`

`a_5 = -1.178259848`

The same method can be applied to find the `4` Complex roots, using Complex arithmetic and Complex initial approximations.

Since my spreadsheet application does not support Complex arithmetic directly, I created columns for `Re(z)`, `Im(z)`, `Re(z^2)`, `Im(z^2)`,.., `Re(z^5)`, `Im(z^5)`, `Re(f(z))`, `Im(f(z))`, `Re(f'(z))`, `Im(f'(z))`, `Re(f(z)bar(f'(z)))`, `Im(f(z)bar(f'(z)))`, `Re(f'(z)bar(f'(z)))`, `Im(f'(z)bar(f'(z)))`.

With an initial approximation of `0+2i` I saw the following approximations:

`a_0 = 0+2i`

`a_1 = -0.13953488372093+1.860465116279i`

`a_2 = -0.21027440054555+1.889317276079i`

`a_3 = -0.20248503148821+1.893034069471i`

`a_4 = -0.20255442138413+1.893126325251i`

`a_5 = -0.20255440359096+1.893126314652i`

`a_6 = -0.20255440359096+1.893126314652i`

With an initial approximation of `1+i` I saw the following approximations:

`a_0 = 1+i`

`a_1 = 0.83921568627451+0.909803921569i`

`a_2 = 0.79458495418768+0.883839764146i`

`a_3 = 0.79169561531938+0.882058297006i`

`a_4 = 0.79168432769581+0.882050595168i`

`a_5 = 0.79168432752598+0.882050595027i`

`a_6 = 0.79168432752598+0.882050595027i`

Note that Complex roots occur in conjugate pairs so the `4` Complex roots are approximately:

`-0.20255440359096+-1.893126314652i`

`0.79168432752598+-0.882050595027i`