How do you find all the zeros of -x^5+3x^4+16x^3-2x^2-95x-44−x5+3x4+16x3−2x2−95x−44?
1 Answer
Use Newton's method to find numeric approximations for the three Real zeros, then divide by the corresponding factors to get a quadratic for the Complex zeros.
Explanation:
f(x) = -x^5+3x^4+16x^3-2x^2-95x-44f(x)=−x5+3x4+16x3−2x2−95x−44
You could try the rational root theorem first, which would allow you to infer that the only possible rational zeros of
+-1±1 ,+-2±2 ,+-4±4 ,+-11±11 ,+-22±22 ,+-44±44 .
None of these work, so
f(-1) = 37f(−1)=37
f(1) = -123f(1)=−123
f(4) = 312f(4)=312
f(11) = -97163f(11)=−97163
So
We can use Newton's method to find good approximations for the Real roots by choosing suitable starting approximations
a_(i+1) = a_i - f(a_i)/(f'(a_i))
In our example,
Putting the iteration formula into a spreadsheet and using initial values
-0.485335316717177
2.624730249302921
5.259365512110042
To find the Complex zeros, you can either put together a more complicated spreadsheet, with separate columns for Real and Imaginary parts, or you can divide
