**Section Objectives**

- Apply the Fundamental Theorem of Algebra.
- Use common tests to determine the nature of a polynomial's zeros.
- Factor a polynomial with real coefficients into a product of linear factors and irreducible quadratic factors.

If is a polynomial with integer coefficients, then any rational zero of must have the form where is a factor of and is a factor of .

- List all possible rational zeros of .

To find the rational zeros of a polynomial with integer coefficients:

- List all rational zeros.
- If appropriate, use a graphing calculator to narrow done the list.
- When very simple to do so, check a possible zero by directly evaluating the polynomial.
- When synthetic division is more appropriate, check a possible zero by using synthetic division to evaluate the polynomial.

- Find all the zeros of . Then give a complete factorization of .

**Descartes' Rule of Signs:** When a polynomial with real coefficients is written is order of decreasing exponents,

- count the sign changes between nonzero coefficients. The number of positive, real zeros is that number or fewer by an even number.
- change the signs of the odd-degree terms and then count the sign changes between nonzero coefficients. The number of negative, real zeros is that number or fewer by an even number.

- Apply Descartes' rules of signs:

- Apply Descartes' rules of signs:

**Intermediate Value Theorem:** If is a polynomial and and are two real numbers for which and have opposite signs, then has at least one real zero between and .

- Show that has at least one real zero between and .

If is a polynomial of degree , , then has at least one zero.

Some important results follow from the fundamental theorem:

- A polynomial of degree has exactly zeros, counting multiplicity.
- A polynomial of degree can be written in factored form as a product of linear factors.
- The graph of a polynomial of degree has at most -intercepts and turning points.

**Conjugate Roots Theorem:** Suppose is a polynomial with real coefficients. If the complex number is a zero of , then the complex number is also a zero of . Thus the complex zeros of a polynomial with real coefficients always occur in conjugate pairs.

A quadratic factor of a polynomial with real coefficients is said to be **irreducible** if its zeros are imaginary complex conjugates.

- Use the quadratic formula to find the zeros of .

- Construct a 4th-degree polynomial with real coefficients with zeros 2, -5, and and whose graph passes through .

Finding all real and complex zeros of a polynomial is typically a difficult task. Here are some guidelines:

- Start by factoring as much as is easily possible.
- Use the quadratic formula if applicable.
- Apply the Rational Zeros Theorem (or other tests).
- Test the possible rational zeros and, each time you find one, use synthetic division to deflate the polynomial (divide out the corresponding linear factor).
- Repeat steps 1-4, as necessary, on the deflated polynomial.
- Use a graphing calculator to help locate the real zeros.
- If necessary, apply more advanced zero-finding techniques.

- Completely factor the polynomial .

- Completely factor the polynomial .