Section 6.8 PART 1Roots and Zeros

By the end of this section, you will be able to:

1. Determine the number and type of roots for a polynomial equation;

2. Find the zeros of a polynomial function.

Class notes from _________________

Assignment Due __________________:

A# 6.81: Page 366 #1-4 all; 11-22 all

1596 – 1650

French mathematician and philosopher

Zeros, Factors, and Roots

Given a polynomial function,

• c is a zero

• x – c is a factor

• c is a root or solution of the polynomial equation

• If c is a real number, then (c, 0) is an intercept on the graph of f(x).

Fundamental Theorem of Algebra

Every polynomial equation with ______________ coordinates and

degree greater than _____________ has at least one ___________ in

the set of _______________ numbers.

Equations can have double, triple, or even quadruple roots. In

general, these are referred to as

________________________________________.

Example 1: Determine Number and Types of Roots Solve each equation. State the number and type of roots.

A. B.

C. D.

Corollary

A polynomial function of _________________ has exactly _______ roots.

Descartes’ Rule of Signs

If is a polynomial with real coefficients, the terms of which are arranged in descending powers of

the variable,

• the number of _____________ real zeros of is the same as the number of _______________ in sign

of the coefficients of the terms, or is less than this by ________________ number, and

• The number of _____________ real zeros of is the same as the number of _______________ in sign

of the coefficients of the terms of or is less than this number by ___________________________.

Example 2: Find numbers of positive and negative zeros

State the possible number of positive real zeros, negative real zeros, and imaginary zeros of

PRACTICE Ex 2: Find numbers of positive and negative zeros

State the possible number of positive real zeros, negative real zeros, and imaginary zeros of