1. Finding Roots of Higher Order Polynomials

2. Quadratics Use factoring when the roots are rational f(x)=x2 – 5x – 14
Use the quadratic formula if it is not factorable or if you’re not sure

3. Roots of Cubic Functions
Find the roots (zeros) of
First, use the table feature of your calculator.
Second, inspect the graph of the function.
Third, choose one of the values you believe to be a zero and use long division by the factor associated with it. If k is a zero, then x – k is a factor.
The quotient will be a quadratic, so use previous methods to find the remaining zeros.

4. Synthetic Division
This is a less cumbersome way to accomplish the same thing as long division.
You must divide by a zero associated with a factor.
Since you begin with the coefficient associated with xn , you must have a coefficient for every power of x from n to zero.
A remainder of zero indicates that the number you divided by is a zero.

5. Rational Roots Theorem
How do you know what to divide by if you can’t use a graphing calculator?
Every rational zero of a function, f, has the form rational zero =
p is a factor of the constant term of the function
q is a factor of the leading coefficient of the function
What are the possible rational zeros of ?

6. Remainders How do you know which of the possible rational zeros work?
If dividing by k gives a remainder of zero, then
k is a zero of the function
(x – k) is a factor of the function
The point (k , 0) is an x-intercept of the function.
If dividing by k does not give a remainder of zero, the remainder is the value of the function at k. That is f(k)=remainder

7. Descartes’ Rule of Signs
Descartes' Rule of Signs will not tell you where the polynomial's zeroes are (you'll need to use the Rational Roots Test and synthetic division, or draw a graph, to actually find the roots), but the Rule will tell you how many roots you can expect.
First, I look at the polynomial as it stands, not changing the sign on x, so this is the "positive" case:
Now I look at f (–x) (that is, having changed the sign on x, so this is the "negative" case):
For detailed information go to:

8. Let’s Practice