Answer the question please with work

Question

Given the function: x^4-5x^3-83x^2-5x-84

a.)  find all zeros (real and complex)

b.) write the function as a pro of linear factors 

Answer 1

Using Descartes' rule of signs, we know that there will be ONE positive Real root and there will be 3 (or 1) negative Real root.

 

Search for a Real root. Since the rational roots theorem requires that any integer zero (root) be a factor of 84, there are MANY choices.  A good way to do this search is to graph the polynomial function on a graphing calculator, with suitable window parameters. Once you find a nice integer x-intercept (or two, you can use synthetic division to find the 'depressed polynomial' after successful synth div by that root (the remainder will be 0). You can do further searcing and synthdiv on each depressed polynomial.

 

Once you have completed all of the above, you'll be able to reconstitute the function in the form of a product of factors, some like (x - root), perhaps one of the form (x²+positive constant), representing one irreducible quadratic factor, the source of two complex conjugate zeros.