4.4The Rational Root

TheoremObjective:

Identify all possible rational roots of a polynomial equation by using the Rational Root Theorem.

Determine the number of positive and negative real roots a polynomial function has.

Possible Rational Roots

Ex: 6 8 7 3 03 2x x x q p

Factors of p: 1 3,

Factors of q: 1 2 3 6, , ,

Possible zeros: p

q

11

2

1

3

1

63

3

2, , , , ,

(12 possible zeros)

Use the graphing calculators to graph the functions and then compare the x-intercepts with the list of possible rational roots.

Be glad it isn’t pre-graphing calculator time. You would have to check each rational solution. The list sometimes get narrowed by the PNI chart, but not always.

Descartes’ Rule of Signs

• If P(x) is a polynomial function whose terms are arranged in descending powers of the variable,– the number of positive real zeros of P(x) is the same

as the number of changes in sign of the coefficients of the terms, or is less than this by an even number, and

– the number of negative real zeros of P(x) is the same as the number of changes in sign of the coefficients of the terms of P(-x), or is less than this by an even number.

Example 1

State the number of positive real zeros, negative real zeros, and imaginary zeros for this function.

g(x) = x4 + x3 + 2x2 – 3x – 1 P N I

Example 2

State the number of positive real zeros, negative real zeros, and imaginary zeros for this function.

r(x) = x10 – x8 + x6 – x4 + x2 – 1 P N I

Example 3

Given a function and one of its zeros, find all the zeros of the function.

f(x) = x3 – 4x2 + 6x – 4; 2

P N I

P N I

31

11

02

Assignment

4.4 Practice Worksheet #1-6, 9-10Do P/Q, PNI charts, and find all roots, not just the rational ones.