1. When given a root and when not given a root
3.6 Finding Roots
When given a root and when not given a root

2. Solving when a zero or factor is not given
The following steps are for solving the problems.
Find possible positive, negative, and imaginary zeros.
Find p/q.
Plug p/q into equation to find zeros. (Substitution, if large factors, try the ones in the middle first).
Use the solution and complete synthetic division( if the degree of the polynomial is greater than 2, repeat steps 3 and 4).
Quadratic Equation, or solve for x.

3. Solving if a root is given
Use either method of division to divide the polynomial by the given solution.
What remains is called a depressed polynomial.
Use factoring or the quadratic formula to solve for the remaining roots.

4. Examples Solve 2x³ + 5x² - 2x -15 x³ - x² - 10x - 8
x³ - x² - 5x -3 given that x-3 is a factor
x³ - 2x² - 5x +6 given that -2 is a solution

5. Creating a polynomial if given solutions or factors
If only given solutions: Put them in factor form (x-solution) and then distribute and simplify
If given the factors: Distribute and simplify.
Example:
Find the polynomial whose solutions are -3, 5, 2