Question

Find the solutions to this equation algebraically using the Fundamental Theorem of Algebra, the Rational Root Theorem, Descartes' Rule of Signs, and the Factor Theorem.
x^3 - 3x^2 - 18x - 176

Answer 1

I doubt anyone will do this whole problem for you, so I will just outline the procedure:

Use the factor theorem to try various factors of 176 (+ or -) and plug them into the function.

When you find some number A such that

A^3-3A^2-18A-176=0

you know that (x-A) is a factor of the polynomial.

You now can synthetically divide the original polynomial by (x-A) and get a new quadratic polynomial, which can (hopefully) be factored.

If not you will have to either repeat the same procedure as before or use the quadratic formula.