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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
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