7.5 roots and zeros of a function

7.5

Roots and Zeros of a Function

Zeros, Factors & RootsZeros, Factors & RootsSummary:Summary:

– cc is a zero of is a zero of f(x)f(x)

– x - cx - c is a factor of is a factor of f(x)f(x)

– cc is a root/solution of is a root/solution of f(x)f(x) = 0 = 0

– If If cc is real, is real, (c,0)(c,0) is an is an x-x-intercept.intercept.

1 2 11 2 1 0 Let ( ) ... .n n

n nf x a x a x a x a x a

Fundamental Theorem of AlgebraFundamental Theorem of Algebra

Every polynomial equation with degree Every polynomial equation with degree greater than zero has at least one greater than zero has at least one

root in the set of complex #’s.root in the set of complex #’s.

An An nnthth degree polynomial equation of the form degree polynomial equation of the form P(x) = 0P(x) = 0

1.1. has exactly has exactly nn roots in the set of complex #’s. roots in the set of complex #’s.

2.2. has exactly has exactly nn zeros. zeros.

State the number and type of roots.State the number and type of roots.Example 1Example 1

2 2 48 0x x 10 0a

Example 2Example 2

State the number and type of roots.State the number and type of roots.Example 3Example 3

4 16 0y 33 18 0a a

Example 4Example 4

Finding # of possible zerosFinding # of possible zeros( ( Descartes’ Rule of Signs)Descartes’ Rule of Signs)

1)Arrange terms of 1)Arrange terms of f(x)f(x) in descending order. in descending order.

2)Find the number of sign changes in 2)Find the number of sign changes in f(x).f(x).Equals the # of positive real zerosEquals the # of positive real zeros

oror

Subtracted by an even # yields the # of positive real zerosSubtracted by an even # yields the # of positive real zeros

3)Find the number of sign changes in 3)Find the number of sign changes in f(-x).f(-x).Equals the # of positive real zerosEquals the # of positive real zeros

oror

Subtracted by an even # yields the # of positive real zerosSubtracted by an even # yields the # of positive real zeros

4) Use a table to record all possibilities4) Use a table to record all possibilities

Ex 5Ex 5

++ -- imagimag totaltotal

Use a table to list ALL possible zeros. 6 3 2( ) 4 2 1p x x x x x

Ex 6Ex 6

++ -- imagimag totaltotal

Use a table to list ALL possible zeros. 4 3 2( ) 3p x x x x x

HomeworkHomework

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