« Division of Polynomials 

A polynomial of degree n has at most n distinct zeros.
Let p(x) be a polynomial function with real coefficients. If a + ib is an imaginary zero of p(x), the conjugate abi is also a zero of p(x).
For a polynomial f(x) and a constant c,
a. If f(c) = 0, then x  c is a factor of f(x).
b. If x  c is a factor of f(x), then f(c) = 0.
The Factor Theorem tells us that if we find a value of c such that f(c) = 0, then x  c is a factor of f(x). And, if x  c is a factor of f(x), then f(c) = 0.
If a polynomial function has integer coefficients, then every rational zero will have the form p/q where p is a factor of the constant and q is a factor of the leading coefficient.
Example 1
Use the Rational Root Test to list all the possible rational zeros for .
Solution:
Step 1: Find factors of the leading coefficient
1, 1, 2, 2, 4, 4
Step 2: Find factors of the constant
1, 1, 2, 2, 5, 5, 10, 10
Step 3: Find all the POSSIBLE rational zeros or roots.
Writing the possible factors as we get:
Here is a final list of all the posible rational zeros, each one written once and reduced:
Example 2
Factor f(x) = into linear factors
Solution
Step 1: Find factors of the leading coefficient
1, 1, 2, 2, 3, 3, 6, 6
Step 2: Find factor of the constant
1, 1, 2, 2, 5, 5, 10, 10
Step 3: Find all the possible rational zeros or roots.
Writing the possible factors as we get:
We check that 5 is the zero of f(x).
Now we use the Factor Theorem
Now we have to solve
The roots are: