Section 5.5

Theorems about Roots of Polynomial Equations

Rational Root Theorem

• All possible rational roots are in the list of:factors of the constantfactors of the L.C.

Example: Any rational root for will be in the list:

3 22x x 2x 5 0

Find all the roots of: ( ) 3 2f x 15x 32x 3x 2

What are the rational roots of:3 22x x 7x 6 0

Conjugate Root Theorem

• A polynomial with rational coefficients will always have complex roots that are complex conjugate pairs. This is also true of irrational roots but only if the coefficients are rational.

• For example, if 2 - 3i is a root then 2 + 3i is also a root. If √5 is a root then -√5 is also a root.

If are roots, what are two other roots?

2 and 1 i

What is a third-degree polynomial function with rational coefficients

that has roots -4 and 2i?