# How do you find all the zeros of #-x^5+3x^4+16x^3-2x^2-95x-44#?

##### 1 Answer

#### 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-44#

You could try the rational root theorem first, which would allow you to infer that the only possible rational zeros of

#+-1# ,#+-2# ,#+-4# ,#+-11# ,#+-22# ,#+-44# .

None of these work, so

#f(-1) = 37#

#f(1) = -123#

#f(4) = 312#

#f(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