7.5 Roots and Zeros of a Function

7.5 Roots and Zeros of a Function

Zeros, Factors & RootsSummary: • c is a zero of f(x) • x - c is a factor of f(x) • c is a root/solution of f(x) = 0 • If c is real, (c,0) is an x-intercept.

Fundamental Theorem of Algebra Every polynomial equation with degree greater than zero has at least one root in the set of complex #’s. An nth degree polynomial equation of the form P(x) = 0 • has exactly n roots in the set of complex #’s. • has exactly n zeros.

State the number and type of roots. Example 2 Example 1 Example 5-1c

State the number and type of roots. Example 4 Example 3 Example 5-1c

Finding # of possible zeros( Descartes’ Rule of Signs) 1)Arrange terms of f(x) in descending order. 2)Find the number of sign changes in f(x). Equals the # of positive real zeros or Subtracted by an even # yields the # of positive real zeros 3)Find the number of sign changes in f(-x). Equals the # of positive real zeros or Subtracted by an even # yields the # of positive real zeros 4) Use a table to record all possibilities

Use a table to list ALL possible zeros. Ex 5 Example 5-2a

Use a table to list ALL possible zeros. Ex 6 Example 5-2a

Homework • Page 375 # 13 – 23 odd

7.5 Roots and Zeros of a Function