« Division of Polynomials

Number of Zeros Theorem

A polynomial of degree n has at most n distinct zeros.

Conjugate Zeros Theorem

Let p(x) be a polynomial function with real coefficients. If a + ib is an imaginary zero of p(x), the conjugate a-bi is also a zero of p(x).

The Factor Theorem

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.

Rational Root Test

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: